SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Bayesian inference</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Bayesian inference</span></span>

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1246091330">

/* start https://en.wikipedia.org/ */

.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"></span></td></tr><tr><td class="sidebar-content">

<a href="Posterior_probability" title="Posterior probability">Posterior</a> = <a href="Likelihood_function" title="Likelihood function">Likelihood</a> × <a href="Prior_probability" title="Prior probability">Prior</a> ÷ <a href="Marginal_likelihood" title="Marginal likelihood">Evidence</a></td>

</tr><tr><th class="sidebar-heading">

Background</th></tr><tr><td class="sidebar-content">

<ul>

<li><a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a></li>

<li><a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a></li>

<li><a href="Bernstein%E2%80%93von_Mises_theorem" title="Bernstein–von Mises theorem">Bernstein–von Mises theorem</a></li>

<li><a href="Coherence_(philosophical_gambling_strategy)" class="mw-redirect" title="Coherence (philosophical gambling strategy)">Coherence</a></li>

<li><a href="Cox's_theorem" title="Cox's theorem">Cox's theorem</a></li>

<li><a href="Cromwell's_rule" title="Cromwell's rule">Cromwell's rule</a></li>

<li><a href="Likelihood_principle" title="Likelihood principle">Likelihood principle</a></li>

<li><a href="Principle_of_indifference" title="Principle of indifference">Principle of indifference</a></li>

<li><a href="Principle_of_maximum_entropy" title="Principle of maximum entropy">Principle of maximum entropy</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Model building</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Conjugate_prior" title="Conjugate prior">Conjugate prior</a></li>

<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Linear regression</a></li>

<li><a href="Empirical_Bayes_method" title="Empirical Bayes method">Empirical Bayes</a></li>

<li><a href="Bayesian_hierarchical_modeling" title="Bayesian hierarchical modeling">Hierarchical model</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Posterior approximation</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a></li>

<li><a href="Laplace's_approximation" title="Laplace's approximation">Laplace's approximation</a></li>

<li><a href="Integrated_nested_Laplace_approximations" title="Integrated nested Laplace approximations">Integrated nested Laplace approximations</a></li>

<li><a href="Variational_Bayesian_methods" title="Variational Bayesian methods">Variational inference</a></li>

<li><a href="Approximate_Bayesian_computation" title="Approximate Bayesian computation">Approximate Bayesian computation</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Estimators</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Bayesian_estimator" class="mw-redirect" title="Bayesian estimator">Bayesian estimator</a></li>

<li><a href="Credible_interval" title="Credible interval">Credible interval</a></li>

<li><a href="Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum a posteriori estimation</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Evidence approximation</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Evidence_lower_bound" title="Evidence lower bound">Evidence lower bound</a></li>

<li><a href="Nested_sampling_algorithm" title="Nested sampling algorithm">Nested sampling</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Model evaluation</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Bayes_factor" title="Bayes factor">Bayes factor</a> (<a href="Bayesian_information_criterion" title="Bayesian information criterion">Schwarz criterion</a>)</li>

<li><a href="Bayesian_model_averaging" class="mw-redirect" title="Bayesian model averaging">Model averaging</a></li>

<li><a href="Posterior_predictive_distribution" title="Posterior predictive distribution">Posterior predictive</a></li></ul></td>

</tr><tr><td class="sidebar-below">

<ul><li><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul></td></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></td></tr></tbody></table>

<p><b>Bayesian inference</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'b' in 'buy'">b</span><span title="/eɪ/: 'a' in 'face'">eɪ</span><span title="'z' in 'zoom'">z</span><span title="/i/: 'y' in 'happy'">i</span><span title="/ə/: 'a' in 'about'">ə</span><span title="'n' in 'nigh'">n</span></span>/</span></span> <i title="English pronunciation respelling"><span style="font-size:90%">BAY</span>-zee-ən</i> or <span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'b' in 'buy'">b</span><span title="/eɪ/: 'a' in 'face'">eɪ</span><span title="/ʒ/: 's' in 'pleasure'">ʒ</span><span title="/ən/: 'on' in 'button'">ən</span></span>/</span></span> <i title="English pronunciation respelling"><span style="font-size:90%">BAY</span>-zhən</i>)<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a method of <a href="Statistical_inference" title="Statistical inference">statistical inference</a> in which <a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a> is used to calculate a probability of a hypothesis, given prior <a href="Evidence" title="Evidence">evidence</a>, and update it as more <a href="Information" title="Information">information</a> becomes available. Fundamentally, Bayesian inference uses a <a href="Prior_probability" title="Prior probability">prior distribution</a> to estimate <a href="Posterior_probability" title="Posterior probability">posterior probabilities.</a> Bayesian inference is an important technique in <a href="Statistics" title="Statistics">statistics</a>, and especially in <a href="Mathematical_statistics" title="Mathematical statistics">mathematical statistics</a>. Bayesian updating is particularly important in the <a href="Sequential_analysis" title="Sequential analysis">dynamic analysis of a sequence of data</a>. Bayesian inference has found application in a wide range of activities, including <a href="Science" title="Science">science</a>, <a href="Engineering" title="Engineering">engineering</a>, <a href="Philosophy" title="Philosophy">philosophy</a>, <a href="Medicine" title="Medicine">medicine</a>, <a href="Sport" title="Sport">sport</a>, and <a href="Law" title="Law">law</a>. In the philosophy of <a href="Decision_theory" title="Decision theory">decision theory</a>, Bayesian inference is closely related to subjective probability, often called "<a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>".

</p>

<div class="mw-heading mw-heading2"><h2 id="Introduction_to_Bayes'_rule">Introduction to Bayes' rule</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a></div>

<div class="mw-heading mw-heading3"><h3 id="Formal_explanation">Formal explanation</h3></div>

<caption><a href="Contingency_table" title="Contingency table">Contingency table</a>

</caption>

<th style="background:var(--background-color-neutral,#eaecf0);color:inherit;background:linear-gradient(to top right,var(--background-color-neutral,#eaecf0) 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,var(--background-color-neutral,#eaecf0) 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">Hypothesis</div><div style="margin-right:2em;text-align:left"><br><br>Evidence</div></th>

<th>Satisfies<br>hypothesis<br><span class="texhtml mvar" style="font-style:italic;">H</span></th>

<th>Violates<br>hypothesis<br><span class="nowrap">⁠<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg H}</annotation>

</semantics>

</math></span><img src="./960718987a946fc39e915198096cecd2eb42b14c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.614ex; height:2.176ex;" alt="{\displaystyle \neg H}" loading="lazy"></span>⁠</span></th>

<th><br>Total

</th></tr>

<tr>

<th>Has evidence<br><span class="texhtml mvar" style="font-style:italic;">E</span>

</th>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H|E)\cdot P(E)}</annotation>

</semantics>

</math></span><img src="./9de334b4a5e87a48238c7a2391b00ec4dded8d23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.05ex; height:2.843ex;" alt="{\displaystyle P(H|E)\cdot P(E)}" loading="lazy"></span><br><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =P(E|H)\cdot P(H)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =P(E|H)\cdot P(H)}</annotation>

</semantics>

</math></span><img src="./41913fda67e0eb348d2645e568cdf22353d5bc4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.792ex; height:2.843ex;" alt="{\displaystyle =P(E|H)\cdot P(H)}" loading="lazy"></span></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\neg H|E)\cdot P(E)}</annotation>

</semantics>

</math></span><img src="./88e8539d1929034eb786d153d8c9ab8f82c2fd9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.601ex; height:2.843ex;" alt="{\displaystyle P(\neg H|E)\cdot P(E)}" loading="lazy"></span><br><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =P(E|\neg H)\cdot P(\neg H)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =P(E|\neg H)\cdot P(\neg H)}</annotation>

</semantics>

</math></span><img src="./10bc00d18c1fc3d89183297ae633281370faf1e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.892ex; height:2.843ex;" alt="{\displaystyle =P(E|\neg H)\cdot P(\neg H)}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E)}</annotation>

</semantics>

</math></span><img src="./687fe7f4688af755503fd00e7538f285e2a9954b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.33ex; height:2.843ex;" alt="{\displaystyle P(E)}" loading="lazy"></span>⁠</span>

</td></tr>

<tr>

<th>No evidence<br><span class="nowrap">⁠<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg E}</annotation>

</semantics>

</math></span><img src="./0d2771dd525bad1bb656104318ae9d6986459164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.326ex; height:2.176ex;" alt="{\displaystyle \neg E}" loading="lazy"></span>⁠</span>

</th>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H|\neg E)\cdot P(\neg E)}</annotation>

</semantics>

</math></span><img src="./c949a9ae84d92df5ded4faa3727f403c3cb56915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.151ex; height:2.843ex;" alt="{\displaystyle P(H|\neg E)\cdot P(\neg E)}" loading="lazy"></span><br><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =P(\neg E|H)\cdot P(H)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =P(\neg E|H)\cdot P(H)}</annotation>

</semantics>

</math></span><img src="./5a7c690f7997a1e4c33d98220dfe1fc9d69dfe97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.342ex; height:2.843ex;" alt="{\displaystyle =P(\neg E|H)\cdot P(H)}" loading="lazy"></span></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\neg H|\neg E)\cdot P(\neg E)}</annotation>

</semantics>

</math></span><img src="./929e7f08934d077319088dd787f918c25cc5d93c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.701ex; height:2.843ex;" alt="{\displaystyle P(\neg H|\neg E)\cdot P(\neg E)}" loading="lazy"></span><br><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =P(\neg E|\neg H)\cdot P(\neg H)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =P(\neg E|\neg H)\cdot P(\neg H)}</annotation>

</semantics>

</math></span><img src="./3c20aa6b77f7a6d2b120d09cd58fd7876ee83a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.443ex; height:2.843ex;" alt="{\displaystyle =P(\neg E|\neg H)\cdot P(\neg H)}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\neg E)}</annotation>

</semantics>

</math></span><img src="./4378b254ef25d13454d3ac96d76a21bd7cd6495b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.881ex; height:2.843ex;" alt="{\displaystyle P(\neg E)}" loading="lazy"></span>=<br><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-P(E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-P(E)}</annotation>

</semantics>

</math></span><img src="./8c88ad9db7f9733ba7a08d3a8da7868e819c3c0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.333ex; height:2.843ex;" alt="{\displaystyle 1-P(E)}" loading="lazy"></span>

</td></tr>

<tr>

</td></tr>

<tr>

<th>Total

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H)}</annotation>

</semantics>

</math></span><img src="./bd7c4e1deccceac8c85d862b60dc64545d67b82e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.618ex; height:2.843ex;" alt="{\displaystyle P(H)}" loading="lazy"></span>⁠</span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\neg H)=1-P(H)}</annotation>

</semantics>

</math></span><img src="./ccda33ec06c1859cfb98c549e1708e2341728025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.888ex; height:2.843ex;" alt="{\displaystyle P(\neg H)=1-P(H)}" loading="lazy"></span></td>

<td style="text-align:center;">1

</td></tr></tbody></table>

<p>Bayesian inference derives the <a href="Posterior_probability" title="Posterior probability">posterior probability</a> as a <a href="Consequence_relation" class="mw-redirect" title="Consequence relation">consequence</a> of two <a href="Antecedent_(logic)" title="Antecedent (logic)">antecedents</a>: a <a href="Prior_probability" title="Prior probability">prior probability</a> and a "<a href="Likelihood_function" title="Likelihood function">likelihood function</a>" derived from a <a href="Statistical_model" title="Statistical model">statistical model</a> for the observed data. Bayesian inference computes the posterior probability according to <a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a>:

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H\mid E)={\frac {P(E\mid H)\cdot P(H)}{P(E)}},}</annotation>

</semantics>

</math></span></span>

where

</p>

<ul><li><span class="texhtml mvar" style="font-style:italic;">H</span> stands for any <i>hypothesis</i> whose probability may be affected by <a href="Experimental_data" title="Experimental data">data</a> (called <i>evidence</i> below). Often there are competing hypotheses, and the task is to determine which is the most probable.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H)}</annotation>

</semantics>

</math></span><img src="./bd7c4e1deccceac8c85d862b60dc64545d67b82e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.618ex; height:2.843ex;" alt="{\displaystyle P(H)}" loading="lazy"></span>, the <i><a href="Prior_probability" title="Prior probability">prior probability</a></i>, is the estimate of the probability of the hypothesis <span class="texhtml mvar" style="font-style:italic;">H</span> <i>before</i> the data <span class="texhtml mvar" style="font-style:italic;">E</span>, the current evidence, is observed.</li>

<li><span class="texhtml mvar" style="font-style:italic;">E</span>, the <i>evidence</i>, corresponds to new data that were not used in computing the prior probability.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H\mid E)}</annotation>

</semantics>

</math></span><img src="./c79731ffa8cc8640d198cf4b474972a2e732c2ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.331ex; height:2.843ex;" alt="{\displaystyle P(H\mid E)}" loading="lazy"></span>, the <i><a href="Posterior_probability" title="Posterior probability">posterior probability</a></i>, is the probability of <span class="texhtml mvar" style="font-style:italic;">H</span> <i>given</i> <span class="texhtml mvar" style="font-style:italic;">E</span>, i.e., <i>after</i> <span class="texhtml mvar" style="font-style:italic;">E</span> is observed. This is what we want to know: the probability of a hypothesis <i>given</i> the observed evidence.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E\mid H)}</annotation>

</semantics>

</math></span><img src="./8a5204605db5c0396b36603bbd3acc267586d0e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.331ex; height:2.843ex;" alt="{\displaystyle P(E\mid H)}" loading="lazy"></span> is the probability of observing <span class="texhtml mvar" style="font-style:italic;">E</span> <i>given</i> <span class="texhtml mvar" style="font-style:italic;">H</span> and is called the <i><a href="Likelihood_function" title="Likelihood function">likelihood</a></i>. As a function of <span class="texhtml mvar" style="font-style:italic;">E</span> with <span class="texhtml mvar" style="font-style:italic;">H</span> fixed, it indicates the compatibility of the evidence with the given hypothesis. The likelihood function is a function of the evidence, <span class="texhtml mvar" style="font-style:italic;">E</span>, while the posterior probability is a function of the hypothesis, <span class="texhtml mvar" style="font-style:italic;">H</span>.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E)}</annotation>

</semantics>

</math></span><img src="./687fe7f4688af755503fd00e7538f285e2a9954b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.33ex; height:2.843ex;" alt="{\displaystyle P(E)}" loading="lazy"></span> is sometimes termed the <a href="Marginal_likelihood" title="Marginal likelihood">marginal likelihood</a> or "model evidence". This factor is the same for all possible hypotheses being considered (as is evident from the fact that the hypothesis <span class="texhtml mvar" style="font-style:italic;">H</span> does not appear anywhere in the symbol, unlike for all the other factors) and hence does not factor into determining the relative probabilities of different hypotheses.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E)>0}</annotation>

</semantics>

</math></span><img src="./2c58247aa8849d270aa1313a621c8c7a22d88143.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.591ex; height:2.843ex;" alt="{\displaystyle P(E)>0}" loading="lazy"></span> (Else one has <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0/0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0/0}</annotation>

</semantics>

</math></span><img src="./f1260a93f6fb76e30f25d5633b42e39e2c2fda79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 0/0}" loading="lazy"></span>.)</li></ul>

<p>For different values of <span class="texhtml mvar" style="font-style:italic;">H</span>, only the factors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H)}</annotation>

</semantics>

</math></span><img src="./bd7c4e1deccceac8c85d862b60dc64545d67b82e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.618ex; height:2.843ex;" alt="{\displaystyle P(H)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(E\mid H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E\mid H)}</annotation>

</semantics>

</math></span><img src="./8a5204605db5c0396b36603bbd3acc267586d0e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.331ex; height:2.843ex;" alt="{\displaystyle P(E\mid H)}" loading="lazy"></span>, both in the numerator, affect the value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(H\mid E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H\mid E)}</annotation>

</semantics>

</math></span><img src="./c79731ffa8cc8640d198cf4b474972a2e732c2ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.331ex; height:2.843ex;" alt="{\displaystyle P(H\mid E)}" loading="lazy"></span> – the posterior probability of a hypothesis is proportional to its prior probability (its inherent likeliness) and the newly acquired likelihood (its compatibility with the new observed evidence).

</p><p>In cases where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg H}</annotation>

</semantics>

</math></span><img src="./960718987a946fc39e915198096cecd2eb42b14c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.614ex; height:2.176ex;" alt="{\displaystyle \neg H}" loading="lazy"></span> ("not <span class="texhtml mvar" style="font-style:italic;">H</span>"), the <a href="Logical_negation" class="mw-redirect" title="Logical negation">logical negation</a> of <span class="texhtml mvar" style="font-style:italic;">H</span>, is a valid likelihood, Bayes' rule can be rewritten as follows:

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P(H\mid E)&={\frac {P(E\mid H)P(H)}{P(E)}}\\\\&={\frac {P(E\mid H)P(H)}{P(E\mid H)P(H)+P(E\mid \neg H)P(\neg H)}}\\\\&={\frac {1}{1+\left({\frac {1}{P(H)}}-1\right){\frac {P(E\mid \neg H)}{P(E\mid H)}}}}\\\end{aligned}}}</annotation>

</semantics>

</math></span></span>

because

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E)=P(E\mid H)P(H)+P(E\mid \neg H)P(\neg H)}</annotation>

</semantics>

</math></span></span>

and

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H)+P(\neg H)=1.}</annotation>

</semantics>

</math></span></span> This focuses attention on the term <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\tfrac {1}{P(H)}}-1\right){\tfrac {P(E\mid \neg H)}{P(E\mid H)}}.}">

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\tfrac {1}{P(H)}}-1\right){\tfrac {P(E\mid \neg H)}{P(E\mid H)}}.}</annotation>

</semantics>

</math></span></span> If that term is approximately 1, then the probability of the hypothesis given the evidence, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(H\mid E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H\mid E)}</annotation>

</semantics>

</math></span><img src="./c79731ffa8cc8640d198cf4b474972a2e732c2ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.331ex; height:2.843ex;" alt="{\displaystyle P(H\mid E)}" loading="lazy"></span>, is about <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation>

</semantics>

</math></span><img src="./edef8290613648790a8ac1a95c2fb7c3972aea2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}" loading="lazy"></span>, about 50% likely - equally likely or not likely. If that term is very small, close to zero, then the probability of the hypothesis, given the evidence, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(H\mid E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H\mid E)}</annotation>

</semantics>

</math></span><img src="./c79731ffa8cc8640d198cf4b474972a2e732c2ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.331ex; height:2.843ex;" alt="{\displaystyle P(H\mid E)}" loading="lazy"></span> is close to 1 or the conditional hypothesis is quite likely. If that term is very large, much larger than 1, then the hypothesis, given the evidence, is quite unlikely. If the hypothesis (without consideration of evidence) is unlikely, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H)}</annotation>

</semantics>

</math></span><img src="./bd7c4e1deccceac8c85d862b60dc64545d67b82e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.618ex; height:2.843ex;" alt="{\displaystyle P(H)}" loading="lazy"></span> is small (but not necessarily astronomically small) and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{P(H)}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{P(H)}}}</annotation>

</semantics>

</math></span><img src="./42e0cad39b69a2e7ef88cf654813d8f4b6353086.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.809ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{P(H)}}}" loading="lazy"></span> is much larger than 1 and this term can be approximated as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {P(E\mid \neg H)}{P(E\mid H)\cdot P(H)}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {P(E\mid \neg H)}{P(E\mid H)\cdot P(H)}}}</annotation>

</semantics>

</math></span><img src="./4c7ec721eabc66d9b4f9c716d1645e13fdfe3482.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.952ex; height:4.843ex;" alt="{\displaystyle {\tfrac {P(E\mid \neg H)}{P(E\mid H)\cdot P(H)}}}" loading="lazy"></span> and relevant probabilities can be compared directly to each other.

</p><p>One quick and easy way to remember the equation would be to use <a href="Conditional_probability#As_an_axiom_of_probability" title="Conditional probability">rule of multiplication</a>:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E\cap H)=P(E\mid H)P(H)=P(H\mid E)P(E).}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Alternatives_to_Bayesian_updating">Alternatives to Bayesian updating</h3></div>

<p>Bayesian updating is widely used and computationally convenient. However, it is not the only updating rule that might be considered rational.

</p><p><a href="Ian_Hacking" title="Ian Hacking">Ian Hacking</a> noted that traditional "<a href="Dutch_book" class="mw-redirect" title="Dutch book">Dutch book</a>" arguments did not specify Bayesian updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch books. Hacking wrote:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> "And neither the Dutch book argument nor any other in the personalist arsenal of proofs of the probability axioms entails the dynamic assumption. Not one entails Bayesianism. So the personalist requires the dynamic assumption to be Bayesian. It is true that in consistency a personalist could abandon the Bayesian model of learning from experience. Salt could lose its savour."

</p><p>Indeed, there are non-Bayesian updating rules that also avoid Dutch books (as discussed in the literature on "<a href="Probability_kinematics" class="mw-redirect" title="Probability kinematics">probability kinematics</a>") following the publication of <a href="Richard_C._Jeffrey" class="mw-redirect" title="Richard C. Jeffrey">Richard C. Jeffrey</a>'s rule, which applies Bayes' rule to the case where the evidence itself is assigned a probability.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The additional hypotheses needed to uniquely require Bayesian updating have been deemed to be substantial, complicated, and unsatisfactory.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Inference_over_exclusive_and_exhaustive_possibilities">Inference over exclusive and exhaustive possibilities</h2></div>

<p>If evidence is simultaneously used to update belief over a set of exclusive and exhaustive propositions, Bayesian inference may be thought of as acting on this belief distribution as a whole.

</p>

<div class="mw-heading mw-heading3"><h3 id="General_formulation">General formulation</h3></div>

<p>Suppose a process is generating independent and identically distributed events <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n},\ n=1,2,3,\ldots }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{n},\ n=1,2,3,\ldots }</annotation>

</semantics>

</math></span><img src="./9d2205791e152b46c2db60cb458ee4dfb7128164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.354ex; height:2.509ex;" alt="{\displaystyle E_{n},\ n=1,2,3,\ldots }" loading="lazy"></span>, but the <a href="Probability_distribution" title="Probability distribution">probability distribution</a> is unknown. Let the event space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> represent the current state of belief for this process. Each model is represented by event <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{m}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{m}}</annotation>

</semantics>

</math></span><img src="./3a6da794af407f26d1eb865a918c28453b32e9b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.929ex; height:2.509ex;" alt="{\displaystyle M_{m}}" loading="lazy"></span>. The conditional probabilities <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(E_{n}\mid M_{m})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E_{n}\mid M_{m})}</annotation>

</semantics>

</math></span><img src="./4a97bce15bade3aa6581a1d713652f9296f4fbc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.355ex; height:2.843ex;" alt="{\displaystyle P(E_{n}\mid M_{m})}" loading="lazy"></span> are specified to define the models. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(M_{m})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(M_{m})}</annotation>

</semantics>

</math></span><img src="./80070c98c4c6d7f32c893462ecf986c61341695b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.484ex; height:2.843ex;" alt="{\displaystyle P(M_{m})}" loading="lazy"></span> is the <a href="Credence_(statistics)" title="Credence (statistics)">degree of belief</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{m}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{m}}</annotation>

</semantics>

</math></span><img src="./3a6da794af407f26d1eb865a918c28453b32e9b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.929ex; height:2.509ex;" alt="{\displaystyle M_{m}}" loading="lazy"></span>. Before the first inference step, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{P(M_{m})\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{P(M_{m})\}}</annotation>

</semantics>

</math></span><img src="./46455ef34ab6727ae171f57b697be773ce545867.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.809ex; height:2.843ex;" alt="{\displaystyle \{P(M_{m})\}}" loading="lazy"></span> is a set of <i>initial prior probabilities</i>. These must sum to 1, but are otherwise arbitrary.

</p><p>Suppose that the process is observed to generate <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\in \{E_{n}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\in \{E_{n}\}}</annotation>

</semantics>

</math></span><img src="./c33fc0a120006f9f052b22946b004664204c1a3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.875ex; height:2.843ex;" alt="{\displaystyle E\in \{E_{n}\}}" loading="lazy"></span>. For each <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\in \{M_{m}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\in \{M_{m}\}}</annotation>

</semantics>

</math></span><img src="./2356659b3e2726fe5df3c0b545d942054bd05355.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.537ex; height:2.843ex;" alt="{\displaystyle M\in \{M_{m}\}}" loading="lazy"></span>, the prior <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(M)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(M)}</annotation>

</semantics>

</math></span><img src="./afa81e21b1d085edd94f267faa1546609999affd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.997ex; height:2.843ex;" alt="{\displaystyle P(M)}" loading="lazy"></span> is updated to the posterior <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(M\mid E)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(M\mid E)}</annotation>

</semantics>

</math></span><img src="./a9188f536504fc7f4d13f1abed8aeba476817df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.71ex; height:2.843ex;" alt="{\displaystyle P(M\mid E)}" loading="lazy"></span>. From <a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a>:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(M\mid E)={\frac {P(E\mid M)}{\sum _{m}{P(E\mid M_{m})P(M_{m})}}}\cdot P(M).}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(M\mid E)={\frac {P(E\mid M)}{\sum _{m}{P(E\mid M_{m})P(M_{m})}}}\cdot P(M).}</annotation>

</semantics>

</math></span></span>

</p><p>Upon observation of further evidence, this procedure may be repeated.

</p>

<div class="mw-heading mw-heading3"><h3 id="Multiple_observations">Multiple observations</h3></div>

<p>For a sequence of <a href="Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a> observations <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} =(e_{1},\dots ,e_{n})}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =(e_{1},\dots ,e_{n})}</annotation>

</semantics>

</math></span><img src="./96c7bfcb6017fa5c6dd82f1804384e503127e519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.283ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} =(e_{1},\dots ,e_{n})}" loading="lazy"></span>, it can be shown by induction that repeated application of the above is equivalent to

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(M\mid \mathbf {E} )={\frac {P(\mathbf {E} \mid M)}{\sum _{m}{P(\mathbf {E} \mid M_{m})P(M_{m})}}}\cdot P(M),}</annotation>

</semantics>

</math></span></span>

where

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\mathbf {E} \mid M)=\prod _{k}{P(e_{k}\mid M)}.}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Parametric_formulation:_motivating_the_formal_description">Parametric formulation: motivating the formal description</h3></div>

<p>By parameterizing the space of models, the belief in all models may be updated in a single step. The distribution of belief over the model space may then be thought of as a distribution of belief over the parameter space. The distributions in this section are expressed as continuous, represented by probability densities, as this is the usual situation. The technique is, however, equally applicable to discrete distributions.

</p><p>Let the vector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\theta }}}</annotation>

</semantics>

</math></span><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"></span> span the parameter space. Let the initial prior distribution over <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\theta }}}</annotation>

</semantics>

</math></span><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"></span> be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }})}</annotation>

</semantics>

</math></span><img src="./268feca73e96ae4149a42153d22da4095356d224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.08ex; height:2.843ex;" alt="{\displaystyle p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }})}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\alpha }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}}</annotation>

</semantics>

</math></span><img src="./a585d2bb19071162720ea56a7b087dab3ec17156.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.769ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\alpha }}}" loading="lazy"></span> is a set of parameters to the prior itself, or <i><a href="Hyperparameter_(Bayesian_statistics)" title="Hyperparameter (Bayesian statistics)">hyperparameters</a></i>. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} =(e_{1},\dots ,e_{n})}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =(e_{1},\dots ,e_{n})}</annotation>

</semantics>

</math></span><img src="./96c7bfcb6017fa5c6dd82f1804384e503127e519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.283ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} =(e_{1},\dots ,e_{n})}" loading="lazy"></span> be a sequence of <a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed</a> event observations, where all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation>

</semantics>

</math></span><img src="./ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"></span> are distributed as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(e\mid {\boldsymbol {\theta }})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(e\mid {\boldsymbol {\theta }})}</annotation>

</semantics>

</math></span><img src="./c228140a2647e23f85a6eea46f94d0da8936eadc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.395ex; height:2.843ex;" alt="{\displaystyle p(e\mid {\boldsymbol {\theta }})}" loading="lazy"></span> for some <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\theta }}}</annotation>

</semantics>

</math></span><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"></span>. <a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a> is applied to find the <a href="Posterior_distribution" class="mw-redirect" title="Posterior distribution">posterior distribution</a> over <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\theta }}}</annotation>

</semantics>

</math></span><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"></span>:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p({\boldsymbol {\theta }}\mid \mathbf {E} ,{\boldsymbol {\alpha }})&={\frac {p(\mathbf {E} \mid {\boldsymbol {\theta }},{\boldsymbol {\alpha }})}{p(\mathbf {E} \mid {\boldsymbol {\alpha }})}}\cdot p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }})\\&={\frac {p(\mathbf {E} \mid {\boldsymbol {\theta }},{\boldsymbol {\alpha }})}{\int p(\mathbf {E} \mid {\boldsymbol {\theta }},{\boldsymbol {\alpha }})p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }})\,d{\boldsymbol {\theta }}}}\cdot p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }}),\end{aligned}}}">

<mtr>

<mtd>

</mrow>

</mrow>

</mrow>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p({\boldsymbol {\theta }}\mid \mathbf {E} ,{\boldsymbol {\alpha }})&={\frac {p(\mathbf {E} \mid {\boldsymbol {\theta }},{\boldsymbol {\alpha }})}{p(\mathbf {E} \mid {\boldsymbol {\alpha }})}}\cdot p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }})\\&={\frac {p(\mathbf {E} \mid {\boldsymbol {\theta }},{\boldsymbol {\alpha }})}{\int p(\mathbf {E} \mid {\boldsymbol {\theta }},{\boldsymbol {\alpha }})p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }})\,d{\boldsymbol {\theta }}}}\cdot p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }}),\end{aligned}}}</annotation>

</semantics>

</math></span></span>

where

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(\mathbf {E} \mid {\boldsymbol {\theta }},{\boldsymbol {\alpha }})=\prod _{k}p(e_{k}\mid {\boldsymbol {\theta }}).}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Formal_description_of_Bayesian_inference">Formal description of Bayesian inference</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definitions">Definitions</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>, a data point in general. This may in fact be a <a href="Random_vector" class="mw-redirect" title="Random vector">vector</a> of values.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span>, the <a href="Parameter" title="Parameter">parameter</a> of the data point's distribution, i.e., <span class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim p(x\mid \theta )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\sim p(x\mid \theta )}</annotation>

</semantics>

</math></span><img src="./e229220de9e183d8e2814fd2b68f495ec64caf56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.764ex; height:2.843ex;" alt="{\displaystyle x\sim p(x\mid \theta )}" loading="lazy"></span>.</span> This may be a <a href="Random_vector" class="mw-redirect" title="Random vector">vector</a> of parameters.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>, the <a href="Hyperparameter_(Bayesian_statistics)" title="Hyperparameter (Bayesian statistics)">hyperparameter</a> of the parameter distribution, i.e., <span class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \sim p(\theta \mid \alpha )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta \sim p(\theta \mid \alpha )}</annotation>

</semantics>

</math></span><img src="./c217fc8ac69afedb83eab8b2b1dd940b2b8eb911.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.683ex; height:2.843ex;" alt="{\displaystyle \theta \sim p(\theta \mid \alpha )}" loading="lazy"></span>.</span> This may be a <a href="Random_vector" class="mw-redirect" title="Random vector">vector</a> of hyperparameters.</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>

</semantics>

</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> is the sample, a set of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> observed data points, i.e., <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\ldots ,x_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{n}}</annotation>

</semantics>

</math></span><img src="./737e02a5fbf8bc31d443c91025339f9fd1de1065.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}}" loading="lazy"></span>.</li>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}}</annotation>

</semantics>

</math></span><img src="./4f5c5435030c952a58a756e691ea64f60c1bd240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\tilde {x}}}" loading="lazy"></span>, a new data point whose distribution is to be predicted.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Bayesian_inference">Bayesian inference</h3></div>

<ul><li>The <a href="Prior_distribution" class="mw-redirect" title="Prior distribution">prior distribution</a> is the distribution of the parameter(s) before any data is observed, i.e. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\theta \mid \alpha )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(\theta \mid \alpha )}</annotation>

</semantics>

</math></span><img src="./e1ba39272e00ae83061a2b1f4c4f3e04345db806.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.583ex; height:2.843ex;" alt="{\displaystyle p(\theta \mid \alpha )}" loading="lazy"></span> . The prior distribution might not be easily determined; in such a case, one possibility may be to use the <a href="Jeffreys_prior" title="Jeffreys prior">Jeffreys prior</a> to obtain a prior distribution before updating it with newer observations.</li>

<li>The <a href="Sampling_distribution" title="Sampling distribution">sampling distribution</a> is the distribution of the observed data conditional on its parameters, i.e. <span class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\mathbf {X} \mid \theta )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(\mathbf {X} \mid \theta )}</annotation>

</semantics>

</math></span><img src="./42011d81f158fafd5cea73226376a1fd9b9435bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.115ex; height:2.843ex;" alt="{\displaystyle p(\mathbf {X} \mid \theta )}" loading="lazy"></span>.</span> This is also termed the <a href="Likelihood_function" title="Likelihood function">likelihood</a>, especially when viewed as a function of the parameter(s), sometimes written <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {L} (\theta \mid \mathbf {X} )=p(\mathbf {X} \mid \theta )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {L} (\theta \mid \mathbf {X} )=p(\mathbf {X} \mid \theta )}</annotation>

</semantics>

</math></span><img src="./6710afe7dee646a3511abe4bb052a2d8bccda007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.433ex; height:2.843ex;" alt="{\displaystyle \operatorname {L} (\theta \mid \mathbf {X} )=p(\mathbf {X} \mid \theta )}" loading="lazy"></span>.</li>

<li>The <a href="Marginal_likelihood" title="Marginal likelihood">marginal likelihood</a> (sometimes also termed the <i>evidence</i>) is the distribution of the observed data <a href="Marginal_distribution" title="Marginal distribution">marginalized</a> over the parameter(s), i.e. <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\mathbf {X} \mid \alpha )=\int p(\mathbf {X} \mid \theta )p(\theta \mid \alpha )d\theta .}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(\mathbf {X} \mid \alpha )=\int p(\mathbf {X} \mid \theta )p(\theta \mid \alpha )d\theta .}</annotation>

</semantics>

</math></span></span> It quantifies the agreement between data and expert opinion, in a geometric sense that can be made precise.<sup id="cite_ref-deCarvalho-Geometry_6-0" class="reference"><a href="#cite_note-deCarvalho-Geometry-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> If the marginal likelihood is 0 then there is no agreement between the data and expert opinion and Bayes' rule cannot be applied.</li>

<li>The <a href="Posterior_distribution" class="mw-redirect" title="Posterior distribution">posterior distribution</a> is the distribution of the parameter(s) after taking into account the observed data. This is determined by <a href="Bayes'_rule" class="mw-redirect" title="Bayes' rule">Bayes' rule</a>, which forms the heart of Bayesian inference: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\theta \mid \mathbf {X} ,\alpha )={\frac {p(\theta ,\mathbf {X} ,\alpha )}{p(\mathbf {X} ,\alpha )}}={\frac {p(\mathbf {X} \mid \theta ,\alpha )p(\theta ,\alpha )}{p(\mathbf {X} \mid \alpha )p(\alpha )}}={\frac {p(\mathbf {X} \mid \theta ,\alpha )p(\theta \mid \alpha )}{p(\mathbf {X} \mid \alpha )}}\propto p(\mathbf {X} \mid \theta ,\alpha )p(\theta \mid \alpha ).}">

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(\theta \mid \mathbf {X} ,\alpha )={\frac {p(\theta ,\mathbf {X} ,\alpha )}{p(\mathbf {X} ,\alpha )}}={\frac {p(\mathbf {X} \mid \theta ,\alpha )p(\theta ,\alpha )}{p(\mathbf {X} \mid \alpha )p(\alpha )}}={\frac {p(\mathbf {X} \mid \theta ,\alpha )p(\theta \mid \alpha )}{p(\mathbf {X} \mid \alpha )}}\propto p(\mathbf {X} \mid \theta ,\alpha )p(\theta \mid \alpha ).}</annotation>

</semantics>

</math></span></span> This is expressed in words as "posterior is proportional to likelihood times prior", or sometimes as "posterior = likelihood times prior, over evidence".</li>

<li>In practice, for almost all complex Bayesian models used in machine learning, the posterior distribution <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\theta \mid \mathbf {X} ,\alpha )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(\theta \mid \mathbf {X} ,\alpha )}</annotation>

</semantics>

</math></span><img src="./800a7ac7cb03d1e58763462ea55d1c6a475ec6ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.637ex; height:2.843ex;" alt="{\displaystyle p(\theta \mid \mathbf {X} ,\alpha )}" loading="lazy"></span> is not obtained in a closed form distribution, mainly because the parameter space for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> can be very high, or the Bayesian model retains certain hierarchical structure formulated from the observations <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>

</semantics>

</math></span><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> and parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span>. In such situations, we need to resort to approximation techniques.<sup id="cite_ref-Lee-GibbsSampler_7-0" class="reference"><a href="#cite_note-Lee-GibbsSampler-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li>

<li>General case: Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{Y}^{x}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{Y}^{x}}</annotation>

</semantics>

</math></span><img src="./d92eb885c1778b36fe0ea7654f5fa17aacfa7d9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.994ex; height:2.843ex;" alt="{\displaystyle P_{Y}^{x}}" loading="lazy"></span> be the conditional distribution of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> given <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=x}</annotation>

</semantics>

</math></span><img src="./0661396d873679039ffe8e908a39f02402d4912d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.408ex; height:2.176ex;" alt="{\displaystyle X=x}" loading="lazy"></span> and let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{X}}</annotation>

</semantics>

</math></span><img src="./8348dd8ce7e6f7f4778ee01fa5bdc7b828afd98c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.125ex; height:2.509ex;" alt="{\displaystyle P_{X}}" loading="lazy"></span> be the distribution of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. The joint distribution is then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{X,Y}(dx,dy)=P_{Y}^{x}(dy)P_{X}(dx)}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{X,Y}(dx,dy)=P_{Y}^{x}(dy)P_{X}(dx)}</annotation>

</semantics>

</math></span><img src="./ec7fe44cedd3314ca245dfd53a2c11e3ed1c9aca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.349ex; height:3.009ex;" alt="{\displaystyle P_{X,Y}(dx,dy)=P_{Y}^{x}(dy)P_{X}(dx)}" loading="lazy"></span>. The conditional distribution <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{X}^{y}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{X}^{y}}</annotation>

</semantics>

</math></span><img src="./40ad7fcb2df4aacda6af4df410c7e2259292cc7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.125ex; height:3.176ex;" alt="{\displaystyle P_{X}^{y}}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> given <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=y}</annotation>

</semantics>

</math></span><img src="./678864c5e9a7ce08acfc22d0d7f726d2cade5b45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.027ex; height:2.509ex;" alt="{\displaystyle Y=y}" loading="lazy"></span> is then determined by</li></ul>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{X}^{y}(A)=E(1_{A}(X)|Y=y)}</annotation>

</semantics>

</math></span></span>Existence and uniqueness of the needed <a href="Conditional_expectation" title="Conditional expectation">conditional expectation</a> is a consequence of the <a href="Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">Radon–Nikodym theorem</a>. This was formulated by <a href="Andrey_Kolmogorov" title="Andrey Kolmogorov">Kolmogorov</a> in his famous book from 1933. Kolmogorov underlines the importance of conditional probability by writing "I wish to call attention to ... and especially the theory of conditional probabilities and conditional expectations ..." in the Preface.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The Bayes theorem determines the posterior distribution from the prior distribution. Uniqueness requires continuity assumptions.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Bayes' theorem can be generalized to include improper prior distributions such as the uniform distribution on the real line.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Modern <a href="Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a> methods have boosted the importance of Bayes' theorem including cases with improper priors.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Bayesian_prediction">Bayesian prediction</h3></div>

<ul><li>The <a href="Posterior_predictive_distribution" title="Posterior predictive distribution">posterior predictive distribution</a> is the distribution of a new data point, marginalized over the posterior: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\tilde {x}}\mid \mathbf {X} ,\alpha )=\int p({\tilde {x}}\mid \theta )p(\theta \mid \mathbf {X} ,\alpha )d\theta }">

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\tilde {x}}\mid \mathbf {X} ,\alpha )=\int p({\tilde {x}}\mid \theta )p(\theta \mid \mathbf {X} ,\alpha )d\theta }</annotation>

</semantics>

</math></span></span></li>

<li>The <a href="Prior_predictive_distribution" class="mw-redirect" title="Prior predictive distribution">prior predictive distribution</a> is the distribution of a new data point, marginalized over the prior: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\tilde {x}}\mid \alpha )=\int p({\tilde {x}}\mid \theta )p(\theta \mid \alpha )d\theta }">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\tilde {x}}\mid \alpha )=\int p({\tilde {x}}\mid \theta )p(\theta \mid \alpha )d\theta }</annotation>

</semantics>

</math></span></span></li></ul>

<p>Bayesian theory calls for the use of the posterior predictive distribution to do <a href="Predictive_inference" class="mw-redirect" title="Predictive inference">predictive inference</a>, i.e., to <a href="Prediction" title="Prediction">predict</a> the distribution of a new, unobserved data point. That is, instead of a fixed point as a prediction, a distribution over possible points is returned. Only this way is the entire posterior distribution of the parameter(s) used. By comparison, prediction in <a href="Frequentist_statistics" class="mw-redirect" title="Frequentist statistics">frequentist statistics</a> often involves finding an optimum point estimate of the parameter(s)—e.g., by <a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> or <a href="Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">maximum a posteriori estimation</a> (MAP)—and then plugging this estimate into the formula for the distribution of a data point. This has the disadvantage that it does not account for any uncertainty in the value of the parameter, and hence will underestimate the <a href="Variance" title="Variance">variance</a> of the predictive distribution.

</p><p>In some instances, frequentist statistics can work around this problem. For example, <a href="Confidence_interval" title="Confidence interval">confidence intervals</a> and <a href="Prediction_interval" title="Prediction interval">prediction intervals</a> in frequentist statistics when constructed from a <a href="Normal_distribution" title="Normal distribution">normal distribution</a> with unknown <a href="Mean" title="Mean">mean</a> and <a href="Variance" title="Variance">variance</a> are constructed using a <a href="Student's_t-distribution" title="Student's t-distribution">Student's t-distribution</a>. This correctly estimates the variance, due to the facts that (1) the average of normally distributed random variables is also normally distributed, and (2) the predictive distribution of a normally distributed data point with unknown mean and variance, using conjugate or uninformative priors, has a Student's t-distribution. In Bayesian statistics, however, the posterior predictive distribution can always be determined exactly—or at least to an arbitrary level of precision when numerical methods are used.

</p><p>Both types of predictive distributions have the form of a <a href="Compound_probability_distribution" title="Compound probability distribution">compound probability distribution</a> (as does the <a href="Marginal_likelihood" title="Marginal likelihood">marginal likelihood</a>). In fact, if the prior distribution is a <a href="Conjugate_prior" title="Conjugate prior">conjugate prior</a>, such that the prior and posterior distributions come from the same family, it can be seen that both prior and posterior predictive distributions also come from the same family of compound distributions. The only difference is that the posterior predictive distribution uses the updated values of the hyperparameters (applying the Bayesian update rules given in the <a href="Conjugate_prior" title="Conjugate prior">conjugate prior</a> article), while the prior predictive distribution uses the values of the hyperparameters that appear in the prior distribution.

</p>

<div class="mw-heading mw-heading2"><h2 id="Mathematical_properties">Mathematical properties</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style>

<div class="mw-heading mw-heading3"><h3 id="Interpretation_of_factor">Interpretation of factor</h3></div>

<p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {P(E\mid M)}{P(E)}}>1\Rightarrow P(E\mid M)>P(E)}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {P(E\mid M)}{P(E)}}>1\Rightarrow P(E\mid M)>P(E)}</annotation>

</semantics>

</math></span><img src="./fefc5adf9f1830e82ea72c4e983defe7ea173aed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.803ex; height:4.843ex;" alt="{\textstyle {\frac {P(E\mid M)}{P(E)}}>1\Rightarrow P(E\mid M)>P(E)}" loading="lazy"></span>. That is, if the model were true, the evidence would be more likely than is predicted by the current state of belief. The reverse applies for a decrease in belief. If the belief does not change, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {P(E\mid M)}{P(E)}}=1\Rightarrow P(E\mid M)=P(E)}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {P(E\mid M)}{P(E)}}=1\Rightarrow P(E\mid M)=P(E)}</annotation>

</semantics>

</math></span><img src="./ca89d04a8e2728a1c0f7346a51ba09254453dbea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.803ex; height:4.843ex;" alt="{\textstyle {\frac {P(E\mid M)}{P(E)}}=1\Rightarrow P(E\mid M)=P(E)}" loading="lazy"></span>. That is, the evidence is independent of the model. If the model were true, the evidence would be exactly as likely as predicted by the current state of belief.

</p>

<div class="mw-heading mw-heading3"><h3 id="Cromwell's_rule">Cromwell's rule</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Cromwell's_rule" title="Cromwell's rule">Cromwell's rule</a></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(M)=0}</annotation>

</semantics>

</math></span><img src="./86c78ea77f5fd2eb0d9f6f515b8692c6d546e452.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.258ex; height:2.843ex;" alt="{\displaystyle P(M)=0}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(M\mid E)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(M\mid E)=0}</annotation>

</semantics>

</math></span><img src="./fe4936caa9d6721395d01420f44e8dc4457591e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.971ex; height:2.843ex;" alt="{\displaystyle P(M\mid E)=0}" loading="lazy"></span>. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(M)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(M)=1}</annotation>

</semantics>

</math></span><img src="./2c9f6299fece5f2b7af3c1efd63a757b590083a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.258ex; height:2.843ex;" alt="{\displaystyle P(M)=1}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(E)>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E)>0}</annotation>

</semantics>

</math></span><img src="./2c58247aa8849d270aa1313a621c8c7a22d88143.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.591ex; height:2.843ex;" alt="{\displaystyle P(E)>0}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(M|E)=1}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(M|E)=1}</annotation>

</semantics>

</math></span><img src="./e4b5eb6223880b7986cb4cc79a8be1ac8be6de8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.68ex; height:2.843ex;" alt="{\displaystyle P(M|E)=1}" loading="lazy"></span>. This can be interpreted to mean that hard convictions are insensitive to counter-evidence.

</p><p>The former follows directly from Bayes' theorem. The latter can be derived by applying the first rule to the event "not <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>" in place of "<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>", yielding "if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-P(M)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-P(M)=0}</annotation>

</semantics>

</math></span><img src="./acb748c826e88ea4aec1ac4410b0419bcfc043f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.261ex; height:2.843ex;" alt="{\displaystyle 1-P(M)=0}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-P(M\mid E)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-P(M\mid E)=0}</annotation>

</semantics>

</math></span><img src="./5ab0d269079eaada48461cb962765c2b0c638f15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.973ex; height:2.843ex;" alt="{\displaystyle 1-P(M\mid E)=0}" loading="lazy"></span>", from which the result immediately follows.

</p>

<div class="mw-heading mw-heading3"><h3 id="Asymptotic_behaviour_of_posterior">Asymptotic behaviour of posterior</h3></div>

<p>Consider the behaviour of a belief distribution as it is updated a large number of times with <a href="Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a> trials. For sufficiently nice prior probabilities, the <a href="Bernstein%E2%80%93von_Mises_theorem" title="Bernstein–von Mises theorem">Bernstein-von Mises theorem</a> gives that in the limit of infinite trials, the posterior converges to a <a href="Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian distribution</a> independent of the initial prior under some conditions firstly outlined and rigorously proven by <a href="Joseph_L._Doob" title="Joseph L. Doob">Joseph L. Doob</a> in 1948, namely if the random variable in consideration has a finite <a href="Probability_space" title="Probability space">probability space</a>. The more general results were obtained later by the statistician <a href="David_A._Freedman_(statistician)" class="mw-redirect" title="David A. Freedman (statistician)">David A. Freedman</a> who published in two seminal research papers in 1963 <sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and 1965 <sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> when and under what circumstances the asymptotic behaviour of posterior is guaranteed. His 1963 paper treats, like Doob (1949), the finite case and comes to a satisfactory conclusion. However, if the random variable has an infinite but countable <a href="Probability_space" title="Probability space">probability space</a> (i.e., corresponding to a die with infinite many faces) the 1965 paper demonstrates that for a dense subset of priors the <a href="Bernstein%E2%80%93von_Mises_theorem" title="Bernstein–von Mises theorem">Bernstein-von Mises theorem</a> is not applicable. In this case there is <a href="Almost_surely" title="Almost surely">almost surely</a> no asymptotic convergence. Later in the 1980s and 1990s <a href="David_A._Freedman_(statistician)" class="mw-redirect" title="David A. Freedman (statistician)">Freedman</a> and <a href="Persi_Diaconis" title="Persi Diaconis">Persi Diaconis</a> continued to work on the case of infinite countable probability spaces.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> To summarise, there may be insufficient trials to suppress the effects of the initial choice, and especially for large (but finite) systems the convergence might be very slow.

</p>

<div class="mw-heading mw-heading3"><h3 id="Conjugate_priors">Conjugate priors</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Conjugate_prior" title="Conjugate prior">Conjugate prior</a></div>

<p>In parameterized form, the prior distribution is often assumed to come from a family of distributions called <a href="Conjugate_prior" title="Conjugate prior">conjugate priors</a>. The usefulness of a conjugate prior is that the corresponding posterior distribution will be in the same family, and the calculation may be expressed in <a href="Closed-form_expression" title="Closed-form expression">closed form</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Estimates_of_parameters_and_predictions">Estimates of parameters and predictions</h3></div>

<p>It is often desired to use a posterior distribution to estimate a parameter or variable. Several methods of Bayesian estimation select <a href="Central_tendency" title="Central tendency">measurements of central tendency</a> from the posterior distribution.

</p><p>For one-dimensional problems, a unique median exists for practical continuous problems. The posterior median is attractive as a <a href="Robust_statistics" title="Robust statistics">robust estimator</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p><p>If there exists a finite mean for the posterior distribution, then the posterior mean is a method of estimation.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {\theta }}=\operatorname {E} [\theta ]=\int \theta \,p(\theta \mid \mathbf {X} ,\alpha )\,d\theta }</annotation>

</semantics>

</math></span></span>

</p><p>Taking a value with the greatest probability defines <a href="Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">maximum <i>a posteriori</i> (MAP)</a> estimates:<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\theta _{\text{MAP}}\}\subset \arg \max _{\theta }p(\theta \mid \mathbf {X} ,\alpha ).}</annotation>

</semantics>

</math></span></span>

</p><p>There are examples where no maximum is attained, in which case the set of MAP estimates is <a href="Empty_set" title="Empty set">empty</a>.

</p><p>There are other methods of estimation that minimize the posterior <i><a href="Risk" title="Risk">risk</a></i> (expected-posterior loss) with respect to a <a href="Loss_function" title="Loss function">loss function</a>, and these are of interest to <a href="Statistical_decision_theory" class="mw-redirect" title="Statistical decision theory">statistical decision theory</a> using the sampling distribution ("frequentist statistics").<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>

</p><p>The <a href="Posterior_predictive_distribution" title="Posterior predictive distribution">posterior predictive distribution</a> of a new observation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {x}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}}</annotation>

</semantics>

</math></span><img src="./4f5c5435030c952a58a756e691ea64f60c1bd240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\tilde {x}}}" loading="lazy"></span> (that is independent of previous observations) is determined by<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\tilde {x}}|\mathbf {X} ,\alpha )=\int p({\tilde {x}},\theta \mid \mathbf {X} ,\alpha )\,d\theta =\int p({\tilde {x}}\mid \theta )p(\theta \mid \mathbf {X} ,\alpha )\,d\theta .}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Probability_of_a_hypothesis">Probability of a hypothesis</h3></div>

<caption><a href="Contingency_table" title="Contingency table">Contingency table</a>

</caption>

<th style="background:var(--background-color-neutral,#eaecf0);color:inherit;background:linear-gradient(to top right,var(--background-color-neutral,#eaecf0) 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,var(--background-color-neutral,#eaecf0) 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">Bowl</div><div style="margin-right:2em;text-align:left"><br>Cookie</div>

</th>

<th><br>Total

</th></tr>

<tr>

<th>Plain, <i>E</i>

</th>

</td></tr>

<tr>

</th>

<td>30

</td></tr>

<tr>

<th>Total

</th>

<td>80

</td></tr>

<tr>

<td colspan="5"><i>P</i>(<i>H</i><sub>1</sub>|<i>E</i>) = 30 / 50 = 0.6

</td></tr></tbody></table>

<p>Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks a bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies. The cookie turns out to be a plain one. How probable is it that Fred picked it out of bowl #1?

</p><p>Intuitively, it seems clear that the answer should be more than a half, since there are more plain cookies in bowl #1. The precise answer is given by Bayes' theorem. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{1}}</annotation>

</semantics>

</math></span><img src="./4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"></span> correspond to bowl #1, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{2}}</annotation>

</semantics>

</math></span><img src="./7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"></span> to bowl #2.

It is given that the bowls are identical from Fred's point of view, thus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(H_{1})=P(H_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H_{1})=P(H_{2})}</annotation>

</semantics>

</math></span><img src="./64b9732a91ab616a55e7edd2e7d3676c65899871.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.179ex; height:2.843ex;" alt="{\displaystyle P(H_{1})=P(H_{2})}" loading="lazy"></span>, and the two must add up to 1, so both are equal to 0.5.

The event <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> is the observation of a plain cookie. From the contents of the bowls, we know that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(E\mid H_{1})=30/40=0.75}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E\mid H_{1})=30/40=0.75}</annotation>

</semantics>

</math></span><img src="./e105aed81081c021c764effa60d3e4b13bdf4816.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.396ex; height:2.843ex;" alt="{\displaystyle P(E\mid H_{1})=30/40=0.75}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(E\mid H_{2})=20/40=0.5.}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E\mid H_{2})=20/40=0.5.}</annotation>

</semantics>

</math></span><img src="./c07c1b90033034b84b7b072263194f3795a6c7f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.881ex; height:2.843ex;" alt="{\displaystyle P(E\mid H_{2})=20/40=0.5.}" loading="lazy"></span> Bayes' formula then yields

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P(H_{1}\mid E)&={\frac {P(E\mid H_{1})\,P(H_{1})}{P(E\mid H_{1})\,P(H_{1})\;+\;P(E\mid H_{2})\,P(H_{2})}}\\\\\ &={\frac {0.75\times 0.5}{0.75\times 0.5+0.5\times 0.5}}\\\\\ &=0.6\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p>Before we observed the cookie, the probability we assigned for Fred having chosen bowl #1 was the prior probability, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(H_{1})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H_{1})}</annotation>

</semantics>

</math></span><img src="./85009a290ad8a188184da807fde697b99331da6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.54ex; height:2.843ex;" alt="{\displaystyle P(H_{1})}" loading="lazy"></span>, which was 0.5. After observing the cookie, we must revise the probability to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(H_{1}\mid E)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(H_{1}\mid E)}</annotation>

</semantics>

</math></span><img src="./2930de0b5ddf2a7e0cd2b0285cb0ea9289aae85a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.253ex; height:2.843ex;" alt="{\displaystyle P(H_{1}\mid E)}" loading="lazy"></span>, which is 0.6.

</p>

<div class="mw-heading mw-heading3"><h3 id="Making_a_prediction">Making a prediction</h3></div>

<p>An archaeologist is working at a site thought to be from the medieval period, between the 11th century to the 16th century. However, it is uncertain exactly when in this period the site was inhabited. Fragments of pottery are found, some of which are glazed and some of which are decorated. It is expected that if the site were inhabited during the early medieval period, then 1% of the pottery would be glazed and 50% of its area decorated, whereas if it had been inhabited in the late medieval period then 81% would be glazed and 5% of its area decorated. How confident can the archaeologist be in the date of inhabitation as fragments are unearthed?

</p><p>The degree of belief in the continuous variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> (century) is to be calculated, with the discrete set of events <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}}</annotation>

</semantics>

</math></span><img src="./3c4b231136f82fdaf1ed4920a8bda80fb850fad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.431ex; height:3.176ex;" alt="{\displaystyle \{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}}" loading="lazy"></span> as evidence. Assuming linear variation of glaze and decoration with time, and that these variables are independent,

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(E=GD\mid C=c)=(0.01+{\frac {0.81-0.01}{16-11}}(c-11))(0.5-{\frac {0.5-0.05}{16-11}}(c-11))}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E=GD\mid C=c)=(0.01+{\frac {0.81-0.01}{16-11}}(c-11))(0.5-{\frac {0.5-0.05}{16-11}}(c-11))}</annotation>

</semantics>

</math></span></span>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E=G{\bar {D}}\mid C=c)=(0.01+{\frac {0.81-0.01}{16-11}}(c-11))(0.5+{\frac {0.5-0.05}{16-11}}(c-11))}</annotation>

</semantics>

</math></span></span>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E={\bar {G}}D\mid C=c)=((1-0.01)-{\frac {0.81-0.01}{16-11}}(c-11))(0.5-{\frac {0.5-0.05}{16-11}}(c-11))}</annotation>

</semantics>

</math></span></span>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(E={\bar {G}}{\bar {D}}\mid C=c)=((1-0.01)-{\frac {0.81-0.01}{16-11}}(c-11))(0.5+{\frac {0.5-0.05}{16-11}}(c-11))}</annotation>

</semantics>

</math></span></span>

</p><p>Assume a uniform prior of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f_{C}(c)=0.2}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f_{C}(c)=0.2}</annotation>

</semantics>

</math></span><img src="./91060d3a706c221a7f2ecfe83ae1058a253b453a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.507ex; height:2.843ex;" alt="{\textstyle f_{C}(c)=0.2}" loading="lazy"></span>, and that trials are <a href="Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a>. When a new fragment of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math></span><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> is discovered, Bayes' theorem is applied to update the degree of belief for each <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span>:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{C}(c\mid E=e)={\frac {P(E=e\mid C=c)}{P(E=e)}}f_{C}(c)={\frac {P(E=e\mid C=c)}{\int _{11}^{16}{P(E=e\mid C=c)f_{C}(c)dc}}}f_{C}(c)}</annotation>

</semantics>

</math></span></span>

</p><p>A computer simulation of the changing belief as 50 fragments are unearthed is shown on the graph. In the simulation, the site was inhabited around 1420, or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=15.2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=15.2}</annotation>

</semantics>

</math></span><img src="./7166b08e33a05e32a75b663f13ea7c6a5e036c95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.24ex; height:2.176ex;" alt="{\displaystyle c=15.2}" loading="lazy"></span>. By calculating the area under the relevant portion of the graph for 50 trials, the archaeologist can say that there is practically no chance the site was inhabited in the 11th and 12th centuries, about 1% chance that it was inhabited during the 13th century, 63% chance during the 14th century and 36% during the 15th century. The <a href="Bernstein%E2%80%93von_Mises_theorem" title="Bernstein–von Mises theorem">Bernstein-von Mises theorem</a> asserts here the asymptotic convergence to the "true" distribution because the <a href="Probability_space" title="Probability space">probability space</a> corresponding to the discrete set of events <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}}</annotation>

</semantics>

</math></span><img src="./3c4b231136f82fdaf1ed4920a8bda80fb850fad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.431ex; height:3.176ex;" alt="{\displaystyle \{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}}" loading="lazy"></span> is finite (see above section on asymptotic behaviour of the posterior).

</p>

<div class="mw-heading mw-heading2"><h2 id="In_frequentist_statistics_and_decision_theory">In frequentist statistics and decision theory</h2></div>

<p>A <a href="Statistical_decision_theory" class="mw-redirect" title="Statistical decision theory">decision-theoretic</a> justification of the use of Bayesian inference was given by <a href="Abraham_Wald" title="Abraham Wald">Abraham Wald</a>, who proved that every unique Bayesian procedure is <a href="Admissible_decision_rule" title="Admissible decision rule">admissible</a>. Conversely, every <a href="Admissible_decision_rule" title="Admissible decision rule">admissible</a> statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures.<sup id="cite_ref-Bickel_&_Doksum_2001,_page_32_20-0" class="reference"><a href="#cite_note-Bickel_&_Doksum_2001,_page_32-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>

</p><p>Wald characterized admissible procedures as Bayesian procedures (and limits of Bayesian procedures), making the Bayesian formalism a central technique in such areas of <a href="Frequentist_inference" title="Frequentist inference">frequentist inference</a> as <a href="Parameter_estimation" class="mw-redirect" title="Parameter estimation">parameter estimation</a>, <a href="Hypothesis_testing" class="mw-redirect" title="Hypothesis testing">hypothesis testing</a>, and computing <a href="Confidence_intervals" class="mw-redirect" title="Confidence intervals">confidence intervals</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> For example:

</p>

<ul><li>"Under some conditions, all admissible procedures are either Bayes procedures or limits of Bayes procedures (in various senses). These remarkable results, at least in their original form, are due essentially to Wald. They are useful because the property of being Bayes is easier to analyze than admissibility."<sup id="cite_ref-Bickel_&_Doksum_2001,_page_32_20-1" class="reference"><a href="#cite_note-Bickel_&_Doksum_2001,_page_32-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li>

<li>"In decision theory, a quite general method for proving admissibility consists in exhibiting a procedure as a unique Bayes solution."<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li>

<li>"In the first chapters of this work, prior distributions with finite support and the corresponding Bayes procedures were used to establish some of the main theorems relating to the comparison of experiments. Bayes procedures with respect to more general prior distributions have played a very important role in the development of statistics, including its asymptotic theory." "There are many problems where a glance at posterior distributions, for suitable priors, yields immediately interesting information. Also, this technique can hardly be avoided in sequential analysis."<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></li>

<li>"A useful fact is that any Bayes decision rule obtained by taking a proper prior over the whole parameter space must be admissible"<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li>

<li>"An important area of investigation in the development of admissibility ideas has been that of conventional sampling-theory procedures, and many interesting results have been obtained."<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Model_selection">Model selection</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Bayesian_model_selection" class="mw-redirect" title="Bayesian model selection">Bayesian model selection</a></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Bayesian_information_criterion" title="Bayesian information criterion">Bayesian information criterion</a></div>

<p>Bayesian methodology also plays a role in <a href="Model_selection" title="Model selection">model selection</a> where the aim is to select one model from a set of competing models that represents most closely the underlying process that generated the observed data. In Bayesian model comparison, the model with the highest <a href="Posterior_probability" title="Posterior probability">posterior probability</a> given the data is selected. The posterior probability of a model depends on the evidence, or <a href="Marginal_likelihood" title="Marginal likelihood">marginal likelihood</a>, which reflects the probability that the data is generated by the model, and on the <a href="Prior_probability" title="Prior probability">prior belief</a> of the model. When two competing models are a priori considered to be equiprobable, the ratio of their posterior probabilities corresponds to the <a href="Bayes_factor" title="Bayes factor">Bayes factor</a>. Since Bayesian model comparison is aimed on selecting the model with the highest posterior probability, this methodology is also referred to as the maximum a posteriori (MAP) selection rule <sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> or the MAP probability rule.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Probabilistic_programming">Probabilistic programming</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Probabilistic_programming" title="Probabilistic programming">Probabilistic programming</a></div>

<p>While conceptually simple, Bayesian methods can be mathematically and numerically challenging. Probabilistic programming languages (PPLs) implement functions to easily build Bayesian models together with efficient automatic inference methods. This helps separate the model building from the inference, allowing practitioners to focus on their specific problems and leaving PPLs to handle the computational details for them.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Statistical_data_analysis">Statistical data analysis</h3></div>

<p>See the separate Wikipedia entry on <a href="Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>, specifically the <a href="Bayesian_statistics#Statistical_modeling" title="Bayesian statistics">statistical modeling</a> section in that page.

</p>

<div class="mw-heading mw-heading3"><h3 id="Computer_applications">Computer applications</h3></div>

<p>Bayesian inference has applications in <a href="Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a> and <a href="Expert_system" title="Expert system">expert systems</a>. Bayesian inference techniques have been a fundamental part of computerized <a href="Pattern_recognition" title="Pattern recognition">pattern recognition</a> techniques since the late 1950s.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> There is also an ever-growing connection between Bayesian methods and simulation-based <a href="Monte_Carlo_method" title="Monte Carlo method">Monte Carlo</a> techniques since complex models cannot be processed in closed form by a Bayesian analysis, while a <a href="Graphical_model" title="Graphical model">graphical model</a> structure <i>may</i> allow for efficient simulation algorithms like the <a href="Gibbs_sampling" title="Gibbs sampling">Gibbs sampling</a> and other <a href="Metropolis%E2%80%93Hastings_algorithm" title="Metropolis–Hastings algorithm">Metropolis–Hastings algorithm</a> schemes.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Recently Bayesian inference has gained popularity among the <a href="Phylogenetics" title="Phylogenetics">phylogenetics</a> community for these reasons; a number of applications allow many demographic and evolutionary parameters to be estimated simultaneously.

</p><p>As applied to <a href="Statistical_classification" title="Statistical classification">statistical classification</a>, Bayesian inference has been used to develop algorithms for identifying <a href="E-mail_spam" class="mw-redirect" title="E-mail spam">e-mail spam</a>. Applications which make use of Bayesian inference for spam filtering include <a href="CRM114_(program)" title="CRM114 (program)">CRM114</a>, DSPAM, <a href="Bogofilter" title="Bogofilter">Bogofilter</a>, <a href="SpamAssassin" class="mw-redirect" title="SpamAssassin">SpamAssassin</a>, <a href="SpamBayes" title="SpamBayes">SpamBayes</a>, <a href="Mozilla" title="Mozilla">Mozilla</a>, XEAMS, and others. Spam classification is treated in more detail in the article on the <a href="Na%C3%AFve_Bayes_classifier" class="mw-redirect" title="Naïve Bayes classifier">naïve Bayes classifier</a>.

</p><p><a href="Solomonoff's_theory_of_inductive_inference" title="Solomonoff's theory of inductive inference">Solomonoff's Inductive inference</a> is the theory of prediction based on observations; for example, predicting the next symbol based upon a given series of symbols. The only assumption is that the environment follows some unknown but computable <a href="Probability_distribution" title="Probability distribution">probability distribution</a>. It is a formal inductive framework that combines two well-studied principles of inductive inference: Bayesian statistics and <a href="Occam's_Razor" class="mw-redirect" title="Occam's Razor">Occam's Razor</a>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Solomonoff's universal prior probability of any prefix <i>p</i> of a computable sequence <i>x</i> is the sum of the probabilities of all programs (for a universal computer) that compute something starting with <i>p</i>. Given some <i>p</i> and any computable but unknown probability distribution from which <i>x</i> is sampled, the universal prior and Bayes' theorem can be used to predict the yet unseen parts of <i>x</i> in optimal fashion.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Bioinformatics_and_healthcare_applications">Bioinformatics and healthcare applications</h3></div>

<p>Bayesian inference has been applied in different <a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a> applications, including differential gene expression analysis.<sup id="cite_ref-:edgr_38-0" class="reference"><a href="#cite_note-:edgr-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Bayesian inference is also used in a general cancer risk model, called <a href="Continuous_Individualized_Risk_Index" title="Continuous Individualized Risk Index">CIRI</a> (Continuous Individualized Risk Index), where serial measurements are incorporated to update a Bayesian model which is primarily built from prior knowledge.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Cosmology_and_astrophysical_applications">Cosmology and astrophysical applications</h3></div>

<p>The Bayesian approach has been central to recent progress in cosmology and astrophysical applications,<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> and extends to a wide range of astrophysical problems, including the characterisation of exoplanet (such as the fitting of atmosphere for <a href="K2-18b" title="K2-18b">k2-18b</a><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup>), parameter constraints with cosmological data,<sup id="cite_ref-ArXiv_1807_44-0" class="reference"><a href="#cite_note-ArXiv_1807-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> and calibration in astrophysical experiments.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup>

</p><p>In cosmology, it is often employed with computational techniques such as <a href="Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a>(MCMC) and <a href="Nested_sampling_algorithm" title="Nested sampling algorithm">Nested sampling algorithm</a> to analyse complex datasets and navigate high-dimensional parameter space. A notable application is to the Planck 2018 CMB data for parameter inference.<sup id="cite_ref-ArXiv_1807_44-1" class="reference"><a href="#cite_note-ArXiv_1807-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup>

The six base cosmological parameters in <a href="Lambda-CDM_model" title="Lambda-CDM model">Lambda-CDM model</a> are not predicted by a theory, but rather fitted from Cosmic microwave background (CMB) data to a chosen model of cosmology (the Lambda-CDM model).<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> The bayesian code for cosmology `cobaya` <sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> sets up cosmological runs and interfaces cosmological likelihoods, Boltzmann code,<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> which computes the predicted CMB anisotropies for any given set of cosmological parameters, with MCMC or nested sampler.

</p><p>This computational framework is not limited to the standard model, it is also essential for testing alternative or extended theories of cosmology, such as theories with early dark energy,<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> or modified gravity theories introducing additional parameters beyond Lambda-CDM. <a href="Bayesian_model_comparison" class="mw-redirect" title="Bayesian model comparison">Bayesian model comparison</a> can then be employed to calculate the evidence for competing models, providing a statistical basis to assess whether the data support them over the standard Lambda-CDM.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="In_the_courtroom">In the courtroom</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Jurimetrics#Bayesian_analysis_of_evidence" title="Jurimetrics">Jurimetrics § Bayesian analysis of evidence</a></div>

<p>Bayesian inference can be used by jurors to coherently accumulate the evidence for and against a defendant, and to see whether, in totality, it meets their personal threshold for "<a href="Beyond_a_reasonable_doubt" class="mw-redirect" title="Beyond a reasonable doubt">beyond a reasonable doubt</a>".<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> Bayes' theorem is applied successively to all evidence presented, with the posterior from one stage becoming the prior for the next. The benefit of a Bayesian approach is that it gives the juror an unbiased, rational mechanism for combining evidence. It may be appropriate to explain Bayes' theorem to jurors in <a href="Bayes'_rule" class="mw-redirect" title="Bayes' rule">odds form</a>, as <a href="Betting_odds" class="mw-redirect" title="Betting odds">betting odds</a> are more widely understood than probabilities. Alternatively, a <a href="Gambling_and_information_theory" title="Gambling and information theory">logarithmic approach</a>, replacing multiplication with addition, might be easier for a jury to handle.

</p>

<p>If the existence of the crime is not in doubt, only the identity of the culprit, it has been suggested that the prior should be uniform over the qualifying population.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> For example, if 1,000 people could have committed the crime, the prior probability of guilt would be 1/1000.

</p><p>The use of Bayes' theorem by jurors is controversial. In the United Kingdom, a defence <a href="Expert_witness" title="Expert witness">expert witness</a> explained Bayes' theorem to the jury in <i><a href="Regina_versus_Denis_John_Adams" class="mw-redirect" title="Regina versus Denis John Adams">R v Adams</a></i>. The jury convicted, but the case went to appeal on the basis that no means of accumulating evidence had been provided for jurors who did not wish to use Bayes' theorem. The Court of Appeal upheld the conviction, but it also gave the opinion that "To introduce Bayes' Theorem, or any similar method, into a criminal trial plunges the jury into inappropriate and unnecessary realms of theory and complexity, deflecting them from their proper task."

</p><p>Gardner-Medwin<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> argues that the criterion on which a verdict in a criminal trial should be based is <i>not</i> the probability of guilt, but rather the <i>probability of the evidence, given that the defendant is innocent</i> (akin to a <a href="Frequentist" class="mw-redirect" title="Frequentist">frequentist</a> <a href="P-value" title="P-value">p-value</a>). He argues that if the posterior probability of guilt is to be computed by Bayes' theorem, the prior probability of guilt must be known. This will depend on the incidence of the crime, which is an unusual piece of evidence to consider in a criminal trial. Consider the following three propositions:

</p>

<dl><dd><i>A</i> – the known facts and testimony could have arisen if the defendant is guilty.</dd>

<dd><i>B</i> – the known facts and testimony could have arisen if the defendant is innocent.</dd>

<dd><i>C</i> – the defendant is guilty.</dd></dl>

<p>Gardner-Medwin argues that the jury should believe both <i>A</i> and not-<i>B</i> in order to convict. <i>A</i> and not-<i>B</i> implies the truth of <i>C</i>, but the reverse is not true. It is possible that <i>B</i> and <i>C</i> are both true, but in this case he argues that a jury should acquit, even though they know that they will be letting some guilty people go free. See also <a href="Lindley's_paradox" title="Lindley's paradox">Lindley's paradox</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Bayesian_epistemology">Bayesian epistemology</h3></div>

<p><a href="Bayesian_epistemology" title="Bayesian epistemology">Bayesian epistemology</a> is a movement that advocates for Bayesian inference as a means of justifying the rules of inductive logic.

</p><p><a href="Karl_Popper" title="Karl Popper">Karl Popper</a> and <a href="David_Miller_(philosopher)" title="David Miller (philosopher)">David Miller</a> have rejected the idea of Bayesian rationalism, i.e. using Bayes rule to make epistemological inferences:<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> It is prone to the same <a href="Vicious_circle" title="Vicious circle">vicious circle</a> as any other <a href="Justificationism" class="mw-redirect" title="Justificationism">justificationist</a> epistemology, because it presupposes what it attempts to justify. According to this view, a rational interpretation of Bayesian inference would see it merely as a probabilistic version of <a href="Falsifiability" title="Falsifiability">falsification</a>, rejecting the belief, commonly held by Bayesians, that high likelihood achieved by a series of Bayesian updates would prove the hypothesis beyond any reasonable doubt, or even with likelihood greater than 0.

</p>

<div class="mw-heading mw-heading3"><h3 id="Other">Other</h3></div>

<ul><li>The <a href="Scientific_method" title="Scientific method">scientific method</a> is sometimes interpreted as an application of Bayesian inference. In this view, Bayes' rule guides (or should guide) the updating of probabilities about <a href="Hypothesis" title="Hypothesis">hypotheses</a> conditional on new observations or <a href="Experiment" title="Experiment">experiments</a>.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> The Bayesian inference has also been applied to treat <a href="Stochastic_scheduling" title="Stochastic scheduling">stochastic scheduling</a> problems with incomplete information by Cai et al. (2009).<sup id="cite_ref-Cai_et_al._2009_59-0" class="reference"><a href="#cite_note-Cai_et_al._2009-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Bayesian_search_theory" title="Bayesian search theory">Bayesian search theory</a> is used to search for lost objects.</li>

<li><a href="Bayesian_inference_in_phylogeny" title="Bayesian inference in phylogeny">Bayesian inference in phylogeny</a></li>

<li><a href="Bayesian_tool_for_methylation_analysis" title="Bayesian tool for methylation analysis">Bayesian tool for methylation analysis</a></li>

<li><a href="Bayesian_approaches_to_brain_function" title="Bayesian approaches to brain function">Bayesian approaches to brain function</a> investigate the brain as a Bayesian mechanism.</li>

<li>Bayesian inference in ecological studies<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup></li>

<li>Bayesian inference is used to estimate parameters in stochastic chemical kinetic models<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup></li>

<li>Bayesian inference in <a href="Econophysics" title="Econophysics">econophysics</a> for currency or prediction of trend changes in financial quotations<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Bayesian_inference_in_marketing" title="Bayesian inference in marketing">Bayesian inference in marketing</a></li>

<li><a href="Bayesian_inference_in_motor_learning" title="Bayesian inference in motor learning">Bayesian inference in motor learning</a></li>

<li>Bayesian inference is used in <a href="Probabilistic_numerics" title="Probabilistic numerics">probabilistic numerics</a> to solve numerical problems</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Bayes_and_Bayesian_inference">Bayes and Bayesian inference</h2></div>

<p>The problem considered by Bayes in Proposition 9 of his essay, "<a href="An_Essay_Towards_Solving_a_Problem_in_the_Doctrine_of_Chances" title="An Essay Towards Solving a Problem in the Doctrine of Chances">An Essay Towards Solving a Problem in the Doctrine of Chances</a>", is the posterior distribution for the parameter <i>a</i> (the success rate) of the <a href="Binomial_distribution" title="Binomial distribution">binomial distribution</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="History_of_statistics#Bayesian_statistics" title="History of statistics">History of statistics § Bayesian statistics</a></div>

<p>The term <i>Bayesian</i> refers to <a href="Thomas_Bayes" title="Thomas Bayes">Thomas Bayes</a> (1701–1761), who proved that probabilistic limits could be placed on an unknown event.<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> However, it was <a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> (1749–1827) who introduced (as Principle VI) what is now called <a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a> and used it to address problems in <a href="Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, medical statistics, <a href="Reliability_(statistics)" title="Reliability (statistics)">reliability</a>, and <a href="Jurisprudence" title="Jurisprudence">jurisprudence</a>.<sup id="cite_ref-Stigler1986_66-0" class="reference"><a href="#cite_note-Stigler1986-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> Early Bayesian inference, which used uniform priors following Laplace's <a href="Principle_of_insufficient_reason" class="mw-redirect" title="Principle of insufficient reason">principle of insufficient reason</a>, was called "<a href="Inverse_probability" title="Inverse probability">inverse probability</a>" (because it <a href="Inductive_reasoning" title="Inductive reasoning">infers</a> backwards from observations to parameters, or from effects to causes<sup id="cite_ref-Fienberg2006_67-0" class="reference"><a href="#cite_note-Fienberg2006-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup>). After the 1920s, "inverse probability" was largely supplanted by a collection of methods that came to be called <a href="Frequentist_statistics" class="mw-redirect" title="Frequentist statistics">frequentist statistics</a>.<sup id="cite_ref-Fienberg2006_67-1" class="reference"><a href="#cite_note-Fienberg2006-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup>

</p><p>In the 20th century, the ideas of Laplace were further developed in two different directions, giving rise to <i>objective</i> and <i>subjective</i> currents in Bayesian practice. In the objective or "non-informative" current, the statistical analysis depends on only the model assumed, the data analyzed,<sup id="cite_ref-Bernardo2005_68-0" class="reference"><a href="#cite_note-Bernardo2005-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> and the method assigning the prior, which differs from one objective Bayesian practitioner to another. In the subjective or "informative" current, the specification of the prior depends on the belief (that is, propositions on which the analysis is prepared to act), which can summarize information from experts, previous studies, etc.

</p><p>In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of <a href="Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a> methods, which removed many of the computational problems, and an increasing interest in nonstandard, complex applications.<sup id="cite_ref-Wolpert2004_69-0" class="reference"><a href="#cite_note-Wolpert2004-69"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> Despite growth of Bayesian research, most undergraduate teaching is still based on frequentist statistics.<sup id="cite_ref-Bernardo2006_70-0" class="reference"><a href="#cite_note-Bernardo2006-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> Nonetheless, Bayesian methods are widely accepted and used, such as for example in the field of <a href="Machine_learning" title="Machine learning">machine learning</a>.<sup id="cite_ref-Bishop2007_71-0" class="reference"><a href="#cite_note-Bishop2007-71"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup>

</p>

/* start https://en.wikipedia.org/ */

.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}

/* end https://en.wikipedia.org/ */

</style><div class="div-col" style="column-width: 30em;">

<ul><li><a href="Bayesian_approaches_to_brain_function" title="Bayesian approaches to brain function">Bayesian approaches to brain function</a></li>

<li><a href="Credibility_theory" title="Credibility theory">Credibility theory</a></li>

<li><a href="Epistemology" title="Epistemology">Epistemology</a></li>

<li><a href="Free_energy_principle" title="Free energy principle">Free energy principle</a></li>

<li><a href="Inductive_probability" title="Inductive probability">Inductive probability</a></li>

<li><a href="Information_field_theory" title="Information field theory">Information field theory</a></li>

<li><a href="Principle_of_maximum_entropy" title="Principle of maximum entropy">Principle of maximum entropy</a></li>

<li><a href="Probabilistic_causation" title="Probabilistic causation">Probabilistic causation</a></li>

<li><a href="Probabilistic_programming" title="Probabilistic programming">Probabilistic programming</a></li></ul></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/Bayesian">"Bayesian"</a>. <i><a href="Merriam-Webster" title="Merriam-Webster">Merriam-Webster.com Dictionary</a></i>. Merriam-Webster.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFHacking1967" class="citation journal cs1">Hacking, Ian (December 1967). "Slightly More Realistic Personal Probability". <i>Philosophy of Science</i>. <b>34</b> (4): 316. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F288169">10.1086/288169</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14344339">14344339</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/bayes-theorem/">"Bayes' Theorem (Stanford Encyclopedia of Philosophy)"</a>. Plato.stanford.edu<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-01-05</span></span>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="Bas_van_Fraassen" title="Bas van Fraassen">van Fraassen, B.</a> (1989) <i>Laws and Symmetry</i>, Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-824860-1</bdi>.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B. (2013). <i>Bayesian Data Analysis</i>, Third Edition. Chapman and Hall/CRC. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4398-4095-5</bdi>.</span>

</li>

<li id="cite_note-deCarvalho-Geometry-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-deCarvalho-Geometry_6-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFde_CarvalhoPageBarney2019" class="citation journal cs1">de Carvalho, Miguel; Page, Garritt; Barney, Bradley (2019). <a rel="nofollow" class="external text" href="https://www.maths.ed.ac.uk/~mdecarv/papers/decarvalho2018.pdf">"On the geometry of Bayesian inference"</a> <span class="cs1-format">(PDF)</span>. <i>Bayesian Analysis</i>. <b>14</b> (4): 1013‒1036. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F18-BA1112">10.1214/18-BA1112</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:88521802">88521802</a>.</cite></span>

</li>

<li id="cite_note-Lee-GibbsSampler-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lee-GibbsSampler_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFLee2021" class="citation journal cs1">Lee, Se Yoon (2021). "Gibbs sampler and coordinate ascent variational inference: A set-theoretical review". <i>Communications in Statistics – Theory and Methods</i>. <b>51</b> (6): <span class="nowrap">1549–</span>1568. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2008.01006">2008.01006</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03610926.2021.1921214">10.1080/03610926.2021.1921214</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220935477">220935477</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFKolmogorov1933" class="citation book cs1">Kolmogorov, A.N. (1933) [1956]. <i>Foundations of the Theory of Probability</i>. Chelsea Publishing Company.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFTjur1980" class="citation book cs1">Tjur, Tue (1980). <a rel="nofollow" class="external text" href="http://archive.org/details/probabilitybased0000tjur"><i>Probability based on Radon measures</i></a>. Internet Archive. Chichester [Eng.]; New York : Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-27824-5</bdi>.</cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFTaraldsenTuftoLindqvist2021" class="citation journal cs1">Taraldsen, Gunnar; Tufto, Jarle; Lindqvist, Bo H. (2021-07-24). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fsjos.12550">"Improper priors and improper posteriors"</a>. <i>Scandinavian Journal of Statistics</i>. <b>49</b> (3): <span class="nowrap">969–</span>991. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fsjos.12550">10.1111/sjos.12550</a></span>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/11250%2F2984409">11250/2984409</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0303-6898">0303-6898</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:237736986">237736986</a>.</cite></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFRobertCasella2004" class="citation book cs1">Robert, Christian P.; Casella, George (2004). <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/1159112760"><i>Monte Carlo Statistical Methods</i></a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1475741452</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1159112760">1159112760</a>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFFreedman1963" class="citation journal cs1">Freedman, DA (1963). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177703871">"On the asymptotic behavior of Bayes' estimates in the discrete case"</a>. <i>The Annals of Mathematical Statistics</i>. <b>34</b> (4): <span class="nowrap">1386–</span>1403. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177703871">10.1214/aoms/1177703871</a></span>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2238346">2238346</a>.</cite></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite id="CITEREFFreedman1965" class="citation journal cs1">Freedman, DA (1965). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177700155">"On the asymptotic behavior of Bayes estimates in the discrete case II"</a>. <i>The Annals of Mathematical Statistics</i>. <b>36</b> (2): <span class="nowrap">454–</span>456. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177700155">10.1214/aoms/1177700155</a></span>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2238150">2238150</a>.</cite></span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><cite id="CITEREFRobinsWasserman2000" class="citation journal cs1">Robins, James; Wasserman, Larry (2000). "Conditioning, likelihood, and coherence: A review of some foundational concepts". <i>Journal of the American Statistical Association</i>. <b>95</b> (452): <span class="nowrap">1340–</span>1346. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.2000.10474344">10.1080/01621459.2000.10474344</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120767108">120767108</a>.</cite></span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFSenKeatingMason1993" class="citation book cs1"><a href="Pranab_K._Sen" title="Pranab K. Sen">Sen, Pranab K.</a>; Keating, J. P.; Mason, R. L. (1993). <i>Pitman's measure of closeness: A comparison of statistical estimators</i>. Philadelphia: SIAM.</cite></span>

</li>

<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite id="CITEREFChoudhuriGhosalRoy2005" class="citation book cs1">Choudhuri, Nidhan; Ghosal, Subhashis; Roy, Anindya (2005-01-01). "Bayesian Methods for Function Estimation". <i>Handbook of Statistics</i>. Bayesian Thinking. Vol. 25. pp. <span class="nowrap">373–</span>414. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.324.3052">10.1.1.324.3052</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0169-7161%2805%2925013-7">10.1016/s0169-7161(05)25013-7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780444515391</bdi>.</cite></span>

</li>

<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.probabilitycourse.com/chapter9/9_1_2_MAP_estimation.php">"Maximum A Posteriori (MAP) Estimation"</a>. <i>www.probabilitycourse.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2017-06-02</span></span>.</cite></span>

</li>

<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><cite id="CITEREFYu" class="citation web cs1">Yu, Angela. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130228060536/http://www.cogsci.ucsd.edu/~ajyu/Teaching/Tutorials/bayes_dt.pdf">"Introduction to Bayesian Decision Theory"</a> <span class="cs1-format">(PDF)</span>. <i>cogsci.ucsd.edu/</i>. Archived from <a rel="nofollow" class="external text" href="http://www.cogsci.ucsd.edu/~ajyu/Teaching/Tutorials/bayes_dt.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2013-02-28.</cite></span>

</li>

<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><cite id="CITEREFHitchcock" class="citation web cs1">Hitchcock, David. <a rel="nofollow" class="external text" href="http://people.stat.sc.edu/Hitchcock/stat535slidesday18.pdf">"Posterior Predictive Distribution Stat Slide"</a> <span class="cs1-format">(PDF)</span>. <i>stat.sc.edu</i>.</cite></span>

</li>

<li id="cite_note-Bickel_&_Doksum_2001,_page_32-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bickel_&_Doksum_2001,_page_32_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bickel_&_Doksum_2001,_page_32_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Bickel & Doksum (2001, p. 32)</span>

</li>

<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><cite id="CITEREFKiefer,_J.Schwartz_R.1965" class="citation journal cs1"><a href="Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer, J.</a>; Schwartz R. (1965). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177700051">"Admissible Bayes Character of T<sup>2</sup>-, R<sup>2</sup>-, and Other Fully Invariant Tests for Multivariate Normal Problems"</a>. <i>Annals of Mathematical Statistics</i>. <b>36</b> (3): <span class="nowrap">747–</span>770. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177700051">10.1214/aoms/1177700051</a></span>.</cite></span>

</li>

<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><cite id="CITEREFSchwartz,_R.1969" class="citation journal cs1">Schwartz, R. (1969). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177697822">"Invariant Proper Bayes Tests for Exponential Families"</a>. <i>Annals of Mathematical Statistics</i>. <b>40</b>: <span class="nowrap">270–</span>283. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177697822">10.1214/aoms/1177697822</a></span>.</cite></span>

</li>

<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><cite id="CITEREFHwang,_J._T.Casella,_George1982" class="citation journal cs1">Hwang, J. T. & Casella, George (1982). <a rel="nofollow" class="external text" href="http://ecommons.cornell.edu/bitstream/1813/32852/1/BU-750-M.pdf">"Minimax Confidence Sets for the Mean of a Multivariate Normal Distribution"</a> <span class="cs1-format">(PDF)</span>. <i>Annals of Statistics</i>. <b>10</b> (3): <span class="nowrap">868–</span>881. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176345877">10.1214/aos/1176345877</a></span>.</cite></span>

</li>

<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><cite id="CITEREFLehmann,_Erich1986" class="citation book cs1"><a href="Erich_Leo_Lehmann" title="Erich Leo Lehmann">Lehmann, Erich</a> (1986). <i>Testing Statistical Hypotheses</i> (Second ed.).</cite> (see p. 309 of Chapter 6.7 "Admissibility", and pp. 17–18 of Chapter 1.8 "Complete Classes"</span>

</li>

<li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><cite id="CITEREFLe_Cam1986" class="citation book cs1"><a href="Lucien_Le_Cam" title="Lucien Le Cam">Le Cam, Lucien</a> (1986). <i>Asymptotic Methods in Statistical Decision Theory</i>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-96307-5</bdi>.</cite> (From "Chapter 12 Posterior Distributions and Bayes Solutions", p. 324)</span>

</li>

<li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><cite id="CITEREFCoxHinkley1974" class="citation book cs1"><a href="David_R._Cox" class="mw-redirect" title="David R. Cox">Cox, D. R.</a>; Hinkley, D.V. (1974). <i>Theoretical Statistics</i>. Chapman and Hall. p. 432. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-04-121537-3</bdi>.</cite></span>

</li>

<li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><cite id="CITEREFCoxHinkley1974" class="citation book cs1"><a href="David_R._Cox" class="mw-redirect" title="David R. Cox">Cox, D. R.</a>; Hinkley, D.V. (1974). <i>Theoretical Statistics</i>. Chapman and Hall. p. 433. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-04-121537-3</bdi>.</cite>)</span>

</li>

<li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><cite id="CITEREFStoicaSelen2004" class="citation journal cs1">Stoica, P.; Selen, Y. (2004). "A review of information criterion rules". <i>IEEE Signal Processing Magazine</i>. <b>21</b> (4): <span class="nowrap">36–</span>47. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FMSP.2004.1311138">10.1109/MSP.2004.1311138</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17338979">17338979</a>.</cite></span>

</li>

<li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><cite id="CITEREFFatermansVan_Aertden_Dekker2019" class="citation journal cs1">Fatermans, J.; Van Aert, S.; den Dekker, A.J. (2019). "The maximum a posteriori probability rule for atom column detection from HAADF STEM images". <i>Ultramicroscopy</i>. <b>201</b>: <span class="nowrap">81–</span>91. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1902.05809">1902.05809</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ultramic.2019.02.003">10.1016/j.ultramic.2019.02.003</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/30991277">30991277</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:104419861">104419861</a>.</cite></span>

</li>

<li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">Bessiere, P., Mazer, E., Ahuactzin, J. M., & Mekhnacha, K. (2013). Bayesian Programming (1 edition) Chapman and Hall/CRC.</span>

</li>

<li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><cite id="CITEREFDaniel_Roy2015" class="citation journal cs1">Daniel Roy (2015). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160110035042/http://probabilistic-programming.org/wiki/Home">"Probabilistic Programming"</a>. <i>probabilistic-programming.org</i>. Archived from <a rel="nofollow" class="external text" href="http://probabilistic-programming.org/wiki/Home">the original</a> on 2016-01-10<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-01-02</span></span>.</cite></span>

</li>

<li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><cite id="CITEREFGhahramani2015" class="citation journal cs1">Ghahramani, Z (2015). <a rel="nofollow" class="external text" href="https://www.repository.cam.ac.uk/handle/1810/248538">"Probabilistic machine learning and artificial intelligence"</a>. <i>Nature</i>. <b>521</b> (7553): <span class="nowrap">452–</span>459. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015Natur.521..452G">2015Natur.521..452G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature14541">10.1038/nature14541</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26017444">26017444</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:216356">216356</a>.</cite></span>

</li>

<li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><cite id="CITEREFFienberg2006" class="citation journal cs1">Fienberg, Stephen E. (2006-03-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F06-BA101">"When did Bayesian inference become "Bayesian"?"</a>. <i>Bayesian Analysis</i>. <b>1</b> (1). <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F06-BA101">10.1214/06-BA101</a></span>.</cite></span>

</li>

<li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><cite id="CITEREFJim_Albert2009" class="citation book cs1">Jim Albert (2009). <i>Bayesian Computation with R, Second edition</i>. New York, Dordrecht, etc.: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-92297-3</bdi>.</cite></span>

</li>

<li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><cite id="CITEREFRathmannerHutterOrmerod2011" class="citation journal cs1">Rathmanner, Samuel; Hutter, Marcus; Ormerod, Thomas C (2011). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fe13061076">"A Philosophical Treatise of Universal Induction"</a>. <i>Entropy</i>. <b>13</b> (6): <span class="nowrap">1076–</span>1136. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1105.5721">1105.5721</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011Entrp..13.1076R">2011Entrp..13.1076R</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fe13061076">10.3390/e13061076</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2499910">2499910</a>.</cite></span>

</li>

<li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><cite id="CITEREFHutterHeOrmerod2007" class="citation journal cs1">Hutter, Marcus; He, Yang-Hui; Ormerod, Thomas C (2007). "On Universal Prediction and Bayesian Confirmation". <i>Theoretical Computer Science</i>. <b>384</b> (2007): <span class="nowrap">33–</span>48. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0709.1516">0709.1516</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007arXiv0709.1516H">2007arXiv0709.1516H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tcs.2007.05.016">10.1016/j.tcs.2007.05.016</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1500830">1500830</a>.</cite></span>

</li>

<li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><cite id="CITEREFGácsVitányi2010" class="citation citeseerx cs1">Gács, Peter; Vitányi, Paul M. B. (2 December 2010). "Raymond J. Solomonoff 1926-2009". <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.186.8268">10.1.1.186.8268</a></span>.</cite></span>

</li>

<li id="cite_note-:edgr-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-:edgr_38-0">^</a></b></span> <span class="reference-text">Robinson, Mark D & McCarthy, Davis J & Smyth, Gordon K edgeR: a Bioconductor package for differential expression analysis of digital gene expression data, Bioinformatics.</span>

</li>

<li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ciri.stanford.edu/">"CIRI"</a>. <i>ciri.stanford.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-11</span></span>.</cite></span>

</li>

<li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><cite id="CITEREFKurtzEsfahaniSchererSoo2019" class="citation journal cs1">Kurtz, David M.; Esfahani, Mohammad S.; Scherer, Florian; Soo, Joanne; Jin, Michael C.; Liu, Chih Long; Newman, Aaron M.; Dührsen, Ulrich; Hüttmann, Andreas (2019-07-25). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7380118">"Dynamic Risk Profiling Using Serial Tumor Biomarkers for Personalized Outcome Prediction"</a>. <i>Cell</i>. <b>178</b> (3): 699–713.e19. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cell.2019.06.011">10.1016/j.cell.2019.06.011</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1097-4172">1097-4172</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7380118">7380118</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/31280963">31280963</a>.</cite></span>

</li>

<li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><cite id="CITEREFTrotta2017" class="citation cs2">Trotta, Roberto (2017-01-05), <a rel="nofollow" class="external text" href="http://arxiv.org/abs/1701.01467"><i>Bayesian Methods in Cosmology</i></a>, arXiv, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.48550%2FarXiv.1701.01467">10.48550/arXiv.1701.01467</a>, arXiv:1701.01467<span class="reference-accessdate">, retrieved <span class="nowrap">2025-07-23</span></span></cite></span>

</li>

<li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><cite id="CITEREFStaicova2025" class="citation cs2">Staicova, Denitsa (2025-02-17), <a rel="nofollow" class="external text" href="http://arxiv.org/abs/2501.06022"><i>Modern Bayesian Sampling Methods for Cosmological Inference: A Comparative Study</i></a>, arXiv, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.48550%2FarXiv.2501.06022">10.48550/arXiv.2501.06022</a>, arXiv:2501.06022<span class="reference-accessdate">, retrieved <span class="nowrap">2025-07-23</span></span></cite></span>

</li>

<li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><cite id="CITEREFMadhusudhanConstantinouHolmbergSarkar2025" class="citation cs2">Madhusudhan, Nikku; Constantinou, Savvas; Holmberg, Måns; Sarkar, Subhajit; Piette, Anjali A. A.; Moses, Julianne I. (2025-04-16), <a rel="nofollow" class="external text" href="http://arxiv.org/abs/2504.12267"><i>New Constraints on DMS and DMDS in the Atmosphere of K2-18 b from JWST MIRI</i></a>, arXiv, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.48550%2FarXiv.2504.12267">10.48550/arXiv.2504.12267</a>, arXiv:2504.12267<span class="reference-accessdate">, retrieved <span class="nowrap">2025-07-23</span></span></cite></span>

</li>

<li id="cite_note-ArXiv_1807-44"><span class="mw-cite-backlink">^ <a href="#cite_ref-ArXiv_1807_44-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ArXiv_1807_44-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFCollaborationAghanimAkramiAshdown2021" class="citation cs2">Collaboration, Planck; Aghanim, N.; Akrami, Y.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A. J.; Barreiro, R. B. (2021-08-09), <a rel="nofollow" class="external text" href="http://arxiv.org/abs/1807.06209"><i>Planck 2018 results. VI. Cosmological parameters</i></a>, arXiv, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.48550%2FarXiv.1807.06209">10.48550/arXiv.1807.06209</a>, arXiv:1807.06209<span class="reference-accessdate">, retrieved <span class="nowrap">2025-07-23</span></span></cite></span>

</li>

<li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><cite id="CITEREFAnsteyAcedoHandley2021" class="citation cs2">Anstey, Dominic; Acedo, Eloy de Lera; Handley, Will (2021-11-11), <a rel="nofollow" class="external text" href="http://arxiv.org/abs/2010.09644"><i>A General Bayesian Framework for Foreground Modelling and Chromaticity Correction for Global 21cm Experiments</i></a>, arXiv, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.48550%2FarXiv.2010.09644">10.48550/arXiv.2010.09644</a>, arXiv:2010.09644<span class="reference-accessdate">, retrieved <span class="nowrap">2025-07-23</span></span></cite></span>

</li>

<li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><cite id="CITEREFLewisBridle2002" class="citation cs2">Lewis, Antony; Bridle, Sarah (2002-10-14), <a rel="nofollow" class="external text" href="http://arxiv.org/abs/astro-ph/0205436"><i>Cosmological parameters from CMB and other data: a Monte-Carlo approach</i></a>, arXiv, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.48550%2FarXiv.astro-ph%2F0205436">10.48550/arXiv.astro-ph/0205436</a>, arXiv:astro-ph/0205436<span class="reference-accessdate">, retrieved <span class="nowrap">2025-07-23</span></span></cite></span>

</li>

<li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://cobaya.readthedocs.io/en/latest/index.html">"Cobaya, a code for Bayesian analysis in Cosmology — cobaya 3.5.7 documentation"</a>. <i>cobaya.readthedocs.io</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2025-07-23</span></span>.</cite></span>

</li>

<li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://camb.readthedocs.io/en/latest/">"CAMB — Code for Anisotropies in the Microwave Background (CAMB) 1.6.1 documentation"</a>. <i>camb.readthedocs.io</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2025-07-23</span></span>.</cite></span>

</li>

<li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><cite id="CITEREFLesgourgues2011" class="citation cs2">Lesgourgues, Julien (2011-05-14), <a rel="nofollow" class="external text" href="http://arxiv.org/abs/1104.2932"><i>The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview</i></a>, arXiv, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.48550%2FarXiv.1104.2932">10.48550/arXiv.1104.2932</a>, arXiv:1104.2932<span class="reference-accessdate">, retrieved <span class="nowrap">2025-07-23</span></span></cite></span>

</li>

<li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><cite id="CITEREFHillMcDonoughToomeyAlexander2020" class="citation cs2">Hill, J. Colin; McDonough, Evan; Toomey, Michael W.; Alexander, Stephon (2020-07-15), <a rel="nofollow" class="external text" href="http://arxiv.org/abs/2003.07355"><i>Early Dark Energy Does Not Restore Cosmological Concordance</i></a>, arXiv, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.48550%2FarXiv.2003.07355">10.48550/arXiv.2003.07355</a>, arXiv:2003.07355<span class="reference-accessdate">, retrieved <span class="nowrap">2025-07-23</span></span></cite></span>

</li>

<li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><cite id="CITEREFTrotta2008" class="citation cs2">Trotta, Roberto (2008-03-28), <a rel="nofollow" class="external text" href="http://arxiv.org/abs/0803.4089"><i>Bayes in the sky: Bayesian inference and model selection in cosmology</i></a>, arXiv, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.48550%2FarXiv.0803.4089">10.48550/arXiv.0803.4089</a>, arXiv:0803.4089<span class="reference-accessdate">, retrieved <span class="nowrap">2025-07-23</span></span></cite></span>

</li>

<li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text">Dawid, A. P. and Mortera, J. (1996) "Coherent Analysis of Forensic Identification Evidence". <i><a href="Journal_of_the_Royal_Statistical_Society" title="Journal of the Royal Statistical Society">Journal of the Royal Statistical Society</a></i>, Series B, 58, 425–443.</span>

</li>

Foreman, L. A.; Smith, A. F. M., and Evett, I. W. (1997). "Bayesian analysis of deoxyribonucleic acid profiling data in forensic identification applications (with discussion)". <i>Journal of the Royal Statistical Society</i>, Series A, 160, 429–469.</span>

</li>

<li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text">Robertson, B. and Vignaux, G. A. (1995) <i>Interpreting Evidence: Evaluating Forensic Science in the Courtroom</i>. John Wiley and Sons. Chichester. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-96026-3</bdi>.</span>

</li>

<li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text">Dawid, A. P. (2001) <a rel="nofollow" class="external text" href="http://128.40.111.250/evidence/content/dawid-paper.pdf">Bayes' Theorem and Weighing Evidence by Juries</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150701112146/http://128.40.111.250/evidence/content/dawid-paper.pdf">Archived</a> 2015-07-01 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span>

</li>

<li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text">Gardner-Medwin, A. (2005) "What Probability Should the Jury Address?". <i><a href="Significance_(journal)" class="mw-redirect" title="Significance (journal)">Significance</a></i>, 2 (1), March 2005.</span>

</li>

<li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><cite id="CITEREFMiller1994" class="citation book cs1">Miller, David (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bh_yCgAAQBAJ"><i>Critical Rationalism</i></a>. Chicago: Open Court. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8126-9197-9</bdi>.</cite></span>

</li>

<li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text">Howson & Urbach (2005), Jaynes (2003)</span>

</li>

<li id="cite_note-Cai_et_al._2009-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cai_et_al._2009_59-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFCaiWuZhou2009" class="citation journal cs1">Cai, X.Q.; Wu, X.Y.; Zhou, X. (2009). "Stochastic scheduling subject to breakdown-repeat breakdowns with incomplete information". <i>Operations Research</i>. <b>57</b> (5): <span class="nowrap">1236–</span>1249. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fopre.1080.0660">10.1287/opre.1080.0660</a>.</cite></span>

</li>

<li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><cite id="CITEREFOgleTuckerCable2014" class="citation journal cs1">Ogle, Kiona; Tucker, Colin; Cable, Jessica M. (2014-01-01). "Beyond simple linear mixing models: process-based isotope partitioning of ecological processes". <i>Ecological Applications</i>. <b>24</b> (1): <span class="nowrap">181–</span>195. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014EcoAp..24..181O">2014EcoAp..24..181O</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1890%2F1051-0761-24.1.181">10.1890/1051-0761-24.1.181</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1939-5582">1939-5582</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24640543">24640543</a>.</cite></span>

</li>

<li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><cite id="CITEREFEvaristoMcDonnellSchollBruijnzeel2016" class="citation journal cs1">Evaristo, Jaivime; McDonnell, Jeffrey J.; Scholl, Martha A.; Bruijnzeel, L. Adrian; Chun, Kwok P. (2016-01-01). "Insights into plant water uptake from xylem-water isotope measurements in two tropical catchments with contrasting moisture conditions". <i>Hydrological Processes</i>. <b>30</b> (18): <span class="nowrap">3210–</span>3227. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016HyPr...30.3210E">2016HyPr...30.3210E</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fhyp.10841">10.1002/hyp.10841</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1099-1085">1099-1085</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:131588159">131588159</a>.</cite></span>

</li>

<li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><cite id="CITEREFGuptaRawlings2014" class="citation journal cs1">Gupta, Ankur; Rawlings, James B. (April 2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4946376">"Comparison of Parameter Estimation Methods in Stochastic Chemical Kinetic Models: Examples in Systems Biology"</a>. <i>AIChE Journal</i>. <b>60</b> (4): <span class="nowrap">1253–</span>1268. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014AIChE..60.1253G">2014AIChE..60.1253G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Faic.14409">10.1002/aic.14409</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-1541">0001-1541</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4946376">4946376</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/27429455">27429455</a>.</cite></span>

</li>

<li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text"><cite id="CITEREFFornalski2016" class="citation journal cs1">Fornalski, K.W. (2016). <a rel="nofollow" class="external text" href="http://www.rroij.com/open-access/the-tadpole-bayesian-model-for-detecting-trend-changesin-financial-quotations-.pdf">"The Tadpole Bayesian Model for Detecting Trend Changes in Financial Quotations"</a> <span class="cs1-format">(PDF)</span>. <i>R&R Journal of Statistics and Mathematical Sciences</i>. <b>2</b> (1): <span class="nowrap">117–</span>122.</cite></span>

</li>

<li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text"><cite id="CITEREFSchützHolschneider2011" class="citation journal cs1">Schütz, N.; Holschneider, M. (2011). "Detection of trend changes in time series using Bayesian inference". <i>Physical Review E</i>. <b>84</b> (2): 021120. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1104.3448">1104.3448</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011PhRvE..84b1120S">2011PhRvE..84b1120S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.84.021120">10.1103/PhysRevE.84.021120</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21928962">21928962</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11460968">11460968</a>.</cite></span>

</li>

<li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text"><cite id="CITEREFStigler1982" class="citation journal cs1">Stigler, Stephen (1982). "Thomas Bayes's Bayesian Inference". <i>Journal of the Royal Statistical Society</i>. <b>145</b> (2): <span class="nowrap">250–</span>58.</cite></span>

</li>

<li id="cite_note-Stigler1986-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stigler1986_66-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFStigler1986" class="citation book cs1">Stigler, Stephen M. (1986). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofstatist00stig">"Chapter 3"</a></span>. <i>The History of Statistics</i>. Harvard University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780674403406</bdi>.</cite></span>

</li>

<li id="cite_note-Fienberg2006-67"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fienberg2006_67-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fienberg2006_67-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFFienberg2006" class="citation journal cs1">Fienberg, Stephen E. (2006). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F06-ba101">"When did Bayesian Inference Become 'Bayesian'?"</a>. <i>Bayesian Analysis</i>. <b>1</b> (1): 1–40 [p. 5]. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F06-ba101">10.1214/06-ba101</a></span>.</cite></span>

</li>

<li id="cite_note-Bernardo2005-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bernardo2005_68-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBernardo2005" class="citation book cs1"><a href="Jos%C3%A9-Miguel_Bernardo" title="José-Miguel Bernardo">Bernardo, José-Miguel</a> (2005). "Reference analysis". <i>Handbook of statistics</i>. Vol. 25. pp. <span class="nowrap">17–</span>90.</cite></span>

</li>

<li id="cite_note-Wolpert2004-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wolpert2004_69-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWolpert2004" class="citation journal cs1">Wolpert, R. L. (2004). "A Conversation with James O. Berger". <i>Statistical Science</i>. <b>19</b> (1): <span class="nowrap">205–</span>218. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.71.6112">10.1.1.71.6112</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F088342304000000053">10.1214/088342304000000053</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2082155">2082155</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120094454">120094454</a>.</cite></span>

</li>

<li id="cite_note-Bernardo2006-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bernardo2006_70-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBernardo2006" class="citation journal cs1"><a href="Jos%C3%A9-Miguel_Bernardo" title="José-Miguel Bernardo">Bernardo, José M.</a> (2006). <a rel="nofollow" class="external text" href="http://www.ime.usp.br/~abe/ICOTS7/Proceedings/PDFs/InvitedPapers/3I2_BERN.pdf">"A Bayesian mathematical statistics primer"</a> <span class="cs1-format">(PDF)</span>. <i>Icots-7</i>.</cite></span>

</li>

<li id="cite_note-Bishop2007-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bishop2007_71-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBishop2007" class="citation book cs1">Bishop, C. M. (2007). <i>Pattern Recognition and Machine Learning</i>. New York: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0387310732</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}

/* end https://en.wikipedia.org/ */

</style><div class="refbegin" style="">

<ul><li>Aster, Richard; Borchers, Brian, and Thurber, Clifford (2012). <i>Parameter Estimation and Inverse Problems</i>, Second Edition, Elsevier. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0123850487</bdi>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0123850485</bdi></li>

<li><cite id="CITEREFBickel,_Peter_J.Doksum,_Kjell_A.2001" class="citation book cs1">Bickel, Peter J. & Doksum, Kjell A. (2001). <i>Mathematical Statistics, Volume 1: Basic and Selected Topics</i> (Second (updated printing 2007) ed.). Pearson Prentice–Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-850363-5</bdi>.</cite></li>

<li><a href="George_E._P._Box" title="George E. P. Box">Box, G. E. P.</a> and <a href="George_Tiao" class="mw-redirect" title="George Tiao">Tiao, G. C.</a> (1973). <i>Bayesian Inference in Statistical Analysis</i>, Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-57428-7</bdi></li>

<li><cite id="CITEREFEdwards,_Ward1968" class="citation book cs1">Edwards, Ward (1968). "Conservatism in Human Information Processing". In Kleinmuntz, B. (ed.). <i>Formal Representation of Human Judgment</i>. Wiley.</cite></li>

<li><cite id="CITEREFEdwards,_Ward1982" class="citation journal cs1">Edwards, Ward (1982). <a href="Daniel_Kahneman" title="Daniel Kahneman">Daniel Kahneman</a>; <a href="Paul_Slovic" title="Paul Slovic">Paul Slovic</a>; <a href="Amos_Tversky" title="Amos Tversky">Amos Tversky</a> (eds.). "Judgment under uncertainty: Heuristics and biases". <i>Science</i>. <b>185</b> (4157): <span class="nowrap">1124–</span>1131. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974Sci...185.1124T">1974Sci...185.1124T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.185.4157.1124">10.1126/science.185.4157.1124</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17835457">17835457</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:143452957">143452957</a>. <q>Chapter: Conservatism in Human Information Processing (excerpted)</q></cite></li>

<li><a href="Edwin_Thompson_Jaynes" title="Edwin Thompson Jaynes">Jaynes E. T.</a> (2003) <i>Probability Theory: The Logic of Science</i>, CUP. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-59271-0</bdi> (<a rel="nofollow" class="external text" href="http://www-biba.inrialpes.fr/Jaynes/prob.html">Link to Fragmentary Edition of March 1996</a>).</li>

<li><cite id="CITEREFHowson,_C.Urbach,_P.2005" class="citation book cs1"><a href="Colin_Howson" title="Colin Howson">Howson, C.</a> & Urbach, P. (2005). <i>Scientific Reasoning: the Bayesian Approach</i> (3rd ed.). <a href="Open_Court_Publishing_Company" title="Open Court Publishing Company">Open Court Publishing Company</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8126-9578-6</bdi>.</cite></li>

<li><cite id="CITEREFPhillipsEdwards2008" class="citation book cs1">Phillips, L. D.; Edwards, Ward (October 2008). "Chapter 6: Conservatism in a Simple Probability Inference Task (<i>Journal of Experimental Psychology</i> (1966) 72: 346-354)". In Jie W. Weiss; David J. Weiss (eds.). <i>A Science of Decision Making:The Legacy of Ward Edwards</i>. Oxford University Press. p. 536. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-532298-9</bdi>.</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>For a full report on the history of Bayesian statistics and the debates with frequentists approaches, read <cite id="CITEREFVallverdu2016" class="citation book cs1">Vallverdu, Jordi (2016). <i>Bayesians Versus Frequentists A Philosophical Debate on Statistical Reasoning</i>. New York: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-662-48638-2</bdi>.</cite></li>

<li><cite id="CITEREFClayton2021" class="citation book cs1">Clayton, Aubrey (August 2021). <a rel="nofollow" class="external text" href="https://cup.columbia.edu/book/bernoullis-fallacy/9780231199940"><i>Bernoulli's Fallacy: Statistical Illogic and the Crisis of Modern Science</i></a>. Columbia University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-231-55335-3</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Elementary">Elementary</h3></div>

<p>The following books are listed in ascending order of probabilistic sophistication:

</p>

<ul><li>Stone, JV (2013), "Bayes' Rule: A Tutorial Introduction to Bayesian Analysis", <a rel="nofollow" class="external text" href="http://jim-stone.staff.shef.ac.uk/BookBayes2012/BayesRuleBookMain.html">Download first chapter here</a>, Sebtel Press, England.</li>

<li><cite id="CITEREFDennis_V._Lindley2013" class="citation book cs1"><a href="Dennis_V._Lindley" class="mw-redirect" title="Dennis V. Lindley">Dennis V. Lindley</a> (2013). <i>Understanding Uncertainty, Revised Edition</i> (2nd ed.). John Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-118-65012-7</bdi>.</cite></li>

<li><cite id="CITEREFColin_HowsonPeter_Urbach2005" class="citation book cs1"><a href="Colin_Howson" title="Colin Howson">Colin Howson</a> & Peter Urbach (2005). <i>Scientific Reasoning: The Bayesian Approach</i> (3rd ed.). <a href="Open_Court_Publishing_Company" title="Open Court Publishing Company">Open Court Publishing Company</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8126-9578-6</bdi>.</cite></li>

<li><cite id="CITEREFBerry,_Donald_A.1996" class="citation book cs1">Berry, Donald A. (1996). <i>Statistics: A Bayesian Perspective</i>. Duxbury. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-23476-8</bdi>.</cite></li>

<li><cite id="CITEREFMorris_H._DeGrootMark_J._Schervish2002" class="citation book cs1"><a href="Morris_H._DeGroot" title="Morris H. DeGroot">Morris H. DeGroot</a> & Mark J. Schervish (2002). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/probabilitystati00degr_0"><i>Probability and Statistics</i></a></span> (third ed.). Addison-Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-52488-8</bdi>.</cite></li>

<li>Bolstad, William M. (2007) <i>Introduction to Bayesian Statistics</i>: Second Edition, John Wiley <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-27020-2</bdi></li>

<li><cite id="CITEREFWinkler,_Robert_L2003" class="citation book cs1">Winkler, Robert L (2003). <i>Introduction to Bayesian Inference and Decision</i> (2nd ed.). Probabilistic. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-9647938-4-2</bdi>.</cite> Updated classic textbook. Bayesian theory clearly presented.</li>

<li>Lee, Peter M. <i>Bayesian Statistics: An Introduction</i>. Fourth Edition (2012), John Wiley <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-1183-3257-3</bdi></li>

<li><cite id="CITEREFCarlin,_Bradley_P.Louis,_Thomas_A.2008" class="citation book cs1">Carlin, Bradley P. & Louis, Thomas A. (2008). <i>Bayesian Methods for Data Analysis, Third Edition</i>. Boca Raton, FL: Chapman and Hall/CRC. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-58488-697-6</bdi>.</cite></li>

<li><cite id="CITEREFGelmanCarlinSternDunson2013" class="citation book cs1"><a href="Andrew_Gelman" title="Andrew Gelman">Gelman, Andrew</a>; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; <a href="Donald_Rubin" title="Donald Rubin">Rubin, Donald B.</a> (2013). <i>Bayesian Data Analysis, Third Edition</i>. Chapman and Hall/CRC. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4398-4095-5</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Intermediate_or_advanced">Intermediate or advanced</h3></div>

<ul><li><cite id="CITEREFBerger,_James_O1985" class="citation book cs1"><a href="James_Berger_(statistician)" class="mw-redirect" title="James Berger (statistician)">Berger, James O</a> (1985). <i>Statistical Decision Theory and Bayesian Analysis</i>. Springer Series in Statistics (Second ed.). Springer-Verlag. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985sdtb.book.....B">1985sdtb.book.....B</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-96098-2</bdi>.</cite></li>

<li><cite id="CITEREFBernardoSmith1994" class="citation book cs1"><a href="Jos%C3%A9-Miguel_Bernardo" title="José-Miguel Bernardo">Bernardo, José M.</a>; <a href="Adrian_Smith_(statistician)" title="Adrian Smith (statistician)">Smith, Adrian F. M.</a> (1994). <i>Bayesian Theory</i>. Wiley.</cite></li>

<li><a href="Morris_H._DeGroot" title="Morris H. DeGroot">DeGroot, Morris H.</a>, <i>Optimal Statistical Decisions</i>. Wiley Classics Library. 2004. (Originally published (1970) by McGraw-Hill.) <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-68029-X</bdi>.</li>

<li><cite id="CITEREFSchervish1995" class="citation book cs1">Schervish, Mark J. (1995). <i>Theory of statistics</i>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94546-0</bdi>.</cite></li>

<li>Jaynes, E. T. (1998). <a rel="nofollow" class="external text" href="http://www-biba.inrialpes.fr/Jaynes/prob.html"><i>Probability Theory: The Logic of Science</i></a>.</li>

<li>O'Hagan, A. and Forster, J. (2003). <i>Kendall's Advanced Theory of Statistics</i>, Volume 2B: <i>Bayesian Inference</i>. Arnold, New York. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-340-52922-9</bdi>.</li>

<li><cite id="CITEREFRobert,_Christian_P2007" class="citation book cs1">Robert, Christian P (2007). <i>The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation</i> (paperback ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-71598-8</bdi>.</cite></li>

<li><a href="Judea_Pearl" title="Judea Pearl">Pearl, Judea</a>. (1988). <i>Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference</i>, San Mateo, CA: Morgan Kaufmann.</li>

<li>Pierre Bessière et al. (2013). "<a rel="nofollow" class="external text" href="http://www.crcpress.com/product/isbn/9781439880326">Bayesian Programming</a>". CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781439880326</bdi></li>

<li>Francisco J. Samaniego (2010). "A Comparison of the Bayesian and Frequentist Approaches to Estimation". Springer. New York, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4419-5940-9</bdi></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Bayesian_approach_to_statistical_problems">"Bayesian approach to statistical problems"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/Bayesian_statistics">Bayesian Statistics</a> from Scholarpedia.</li>

<li><a rel="nofollow" class="external text" href="http://www.dcs.qmw.ac.uk/%7Enorman/BBNs/BBNs.htm">Introduction to Bayesian probability</a> from Queen Mary University of London</li>

<li><a rel="nofollow" class="external text" href="http://webuser.bus.umich.edu/plenk/downloads.htm">Mathematical Notes on Bayesian Statistics and Markov Chain Monte Carlo</a></li>

<li><a rel="nofollow" class="external text" href="http://cocosci.berkeley.edu/tom/bayes.html">Bayesian reading list</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110625052506/http://cocosci.berkeley.edu/tom/bayes.html">Archived</a> 2011-06-25 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, categorized and annotated by <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060711151352/http://psychology.berkeley.edu/faculty/profiles/tgriffiths.html">Tom Griffiths</a></li>

<li>A. Hajek and S. Hartmann: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110728055439/http://stephanhartmann.org/HajekHartmann_BayesEpist.pdf">Bayesian Epistemology</a>, in: J. Dancy et al. (eds.), A Companion to Epistemology. Oxford: Blackwell 2010, 93–106.</li>

<li>S. Hartmann and J. Sprenger: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110728055519/http://stephanhartmann.org/HartmannSprenger_BayesEpis.pdf">Bayesian Epistemology</a>, in: S. Bernecker and D. Pritchard (eds.), Routledge Companion to Epistemology. London: Routledge 2010, 609–620.</li>

<li><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-inductive/"><i>Stanford Encyclopedia of Philosophy</i>: "Inductive Logic"</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150905093734/http://faculty-staff.ou.edu/H/James.A.Hawthorne-1/Hawthorne--Bayesian_Confirmation_Theory.pdf">Bayesian Confirmation Theory</a> (PDF)</li>

<li><a rel="nofollow" class="external text" href="http://www.faqs.org/faqs/ai-faq/neural-nets/part3/section-7.html">What is Bayesian Learning?</a></li>

<li><a rel="nofollow" class="external text" href="https://causascientia.org/math_stat/DataUnkInf.html"><i>Data, Uncertainty and Inference</i></a> — Informal introduction with many examples, ebook (PDF) freely available at <a rel="nofollow" class="external text" href="https://causascientia.org">causaScientia</a></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Statistics654" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistics654" style="font-size:114%;margin:0 4em"><a href="Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="Outline_of_statistics" title="Outline of statistics">Outline</a></li>

<li><a href="List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics654" style="font-size:114%;margin:0 4em"><a href="Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li>

<li><a href="Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li>

<li><a href="Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li>

<li><a href="Cubic_mean" title="Cubic mean">Cubic</a></li>

<li><a href="Generalized_mean" title="Generalized mean">Generalized/power</a></li>

<li><a href="Geometric_mean" title="Geometric mean">Geometric</a></li>

<li><a href="Harmonic_mean" title="Harmonic mean">Harmonic</a></li>

<li><a href="Heronian_mean" title="Heronian mean">Heronian</a></li>

<li><a href="Heinz_mean" title="Heinz mean">Heinz</a></li>

<li><a href="Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li>

<li><a href="Median" title="Median">Median</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li>

<li><a href="Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li>

<li><a href="Interquartile_range" title="Interquartile range">Interquartile range</a></li>

<li><a href="Percentile" title="Percentile">Percentile</a></li>

<li><a href="Range_(statistics)" title="Range (statistics)">Range</a></li>

<li><a href="Standard_deviation" title="Standard deviation">Standard deviation</a></li>

<li><a href="Variance#Sample_variance" title="Variance">Variance</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Moment_(mathematics)" title="Moment (mathematics)">Moments</a>

<ul><li><a href="Kurtosis" title="Kurtosis">Kurtosis</a></li>

<li><a href="L-moment" title="L-moment">L-moments</a></li>

<li><a href="Skewness" title="Skewness">Skewness</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li>

<li><a href="Grouped_data" title="Grouped data">Grouped data</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li>

<li><a href="Rank_correlation" title="Rank correlation">Rank correlation</a>

<ul><li><a href="Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li>

<li><a href="Spearman's_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bar_chart" title="Bar chart">Bar chart</a></li>

<li><a href="Biplot" title="Biplot">Biplot</a></li>

<li><a href="Control_chart" title="Control chart">Control chart</a></li>

<li><a href="Correlogram" title="Correlogram">Correlogram</a></li>

<li><a href="Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li>

<li><a href="Forest_plot" title="Forest plot">Forest plot</a></li>

<li><a href="Histogram" title="Histogram">Histogram</a></li>

<li><a href="Pie_chart" title="Pie chart">Pie chart</a></li>

<li><a href="Radar_chart" title="Radar chart">Radar chart</a></li>

<li><a href="Run_chart" title="Run chart">Run chart</a></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li>

<li><a href="Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li>

<li><a href="Violin_plot" title="Violin plot">Violin plot</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection654" style="font-size:114%;margin:0 4em"><a href="Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Effect_size" title="Effect size">Effect size</a></li>

<li><a href="Missing_data" title="Missing data">Missing data</a></li>

<li><a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li>

<li><a href="Statistical_population" title="Statistical population">Population</a></li>

<li><a href="Replication_(statistics)" title="Replication (statistics)">Replication</a></li>

<li><a href="Sample_size_determination" title="Sample size determination">Sample size determination</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sampling_(statistics)" title="Sampling (statistics)">Sampling</a>

<ul><li><a href="Cluster_sampling" title="Cluster sampling">Cluster</a></li>

<li><a href="Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li>

<li><a href="Opinion_poll" title="Opinion poll">Opinion poll</a></li>

<li><a href="Questionnaire" title="Questionnaire">Questionnaire</a></li>

<li><a href="Standard_error" title="Standard error">Standard error</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li>

<li><a href="Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li>

<li><a href="Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li>

<li><a href="Random_assignment" title="Random assignment">Random assignment</a></li>

<li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li>

<li><a href="Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li>

<li><a href="Scientific_control" title="Scientific control">Scientific control</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li>

<li><a href="Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li>

<li><a href="Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohort_study" title="Cohort study">Cohort study</a></li>

<li><a href="Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li>

<li><a href="Natural_experiment" title="Natural experiment">Natural experiment</a></li>

<li><a href="Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference654" style="font-size:114%;margin:0 4em"><a href="Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a></li>

<li><a href="Sampling_distribution" title="Sampling distribution">Sampling distribution</a>

<ul><li><a href="Order_statistic" title="Order statistic">Order statistic</a></li></ul></li>

<li><a href="Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a>

<ul><li><a href="Density_estimation" title="Density estimation">Density estimation</a></li></ul></li>

<li><a href="Statistical_model" title="Statistical model">Statistical model</a>

<ul><li><a href="Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li>

<li><a href="Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li>

<li><a href="Statistical_parameter" title="Statistical parameter">Parameter</a>

<ul><li><a href="Location_parameter" title="Location parameter">location</a></li>

<li><a href="Scale_parameter" title="Scale parameter">scale</a></li>

<li><a href="Shape_parameter" title="Shape parameter">shape</a></li></ul></li>

<li><a href="Parametric_statistics" title="Parametric statistics">Parametric family</a>

<ul><li><a href="Likelihood_function" title="Likelihood function">Likelihood</a> <a href="Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size: 85%;">(monotone)</span></a></li>

<li><a href="Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li>

<li><a href="Exponential_family" title="Exponential family">Exponential family</a></li></ul></li>

<li><a href="Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li>

<li><a href="Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li>

<li><a href="Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Optimal_decision" title="Optimal decision">Optimal decision</a>

<ul><li><a href="Loss_function" title="Loss function">loss function</a></li></ul></li>

<li><a href="Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li>

<li><a href="Statistical_distance" title="Statistical distance">Statistical distance</a>

<ul><li><a href="Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li>

<li><a href="Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li>

<li><a href="Robust_statistics" title="Robust statistics">Robustness</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Estimating_equations" title="Estimating equations">Estimating equations</a>

<ul><li><a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li>

<li><a href="Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li>

<li><a href="M-estimator" title="M-estimator">M-estimator</a></li>

<li><a href="Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li>

<li><a href="Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a>

<ul><li><a href="Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a>

<ul><li><a href="Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li>

<li><a href="Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li>

<li><a href="Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Confidence_interval" title="Confidence interval">Confidence interval</a></li>

<li><a href="Pivotal_quantity" title="Pivotal quantity">Pivot</a></li>

<li><a href="Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li>

<li><a href="Prediction_interval" title="Prediction interval">Prediction interval</a></li>

<li><a href="Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li>

<li><a href="Resampling_(statistics)" title="Resampling (statistics)">Resampling</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li>

<li><a href="Power_(statistics)" title="Power (statistics)">Power</a>

<ul><li><a href="Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li>

<li><a href="Permutation_test" title="Permutation test">Permutation test</a>

<ul><li><a href="Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li>

<li><a href="Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li>

<li><a href="Score_test" title="Score test">Score/Lagrange multiplier</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Z-test" title="Z-test"><i>Z</i>-test <span style="font-size: 85%;">(normal)</span></a></li>

<li><a href="Student's_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chi-squared_test" title="Chi-squared test">Chi-squared</a></li>

<li><a href="Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li>

<li><a href="Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li>

<li><a href="Lilliefors_test" title="Lilliefors test">Lilliefors</a></li>

<li><a href="Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li>

<li><a href="Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size: 85%;">(Shapiro–Wilk)</span></a></li>

<li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li>

<li><a href="Model_selection" title="Model selection">Model selection</a>

<ul><li><a href="Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li>

<li><a href="Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size: 85%;">(Wilcoxon)</span></a>

<ul><li><a href="Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li>

<li><a href="Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size: 85%;">(Mann–Whitney)</span></a></li>

<li><a href="Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="Analysis_of_variance" title="Analysis of variance">anova</a>

<ul><li><a href="Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size: 85%;">(Kruskal–Wallis)</span></a></li>

<li><a href="Friedman_test" title="Friedman test">2-way <span style="font-size: 85%;">(Friedman)</span></a></li>

<li><a href="Jonckheere's_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size: 85%;">(Jonckheere–Terpstra)</span></a></li></ul></li>

<li><a href="Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>

<ul><li><a href="Prior_probability" title="Prior probability">prior</a></li>

<li><a href="Posterior_probability" title="Posterior probability">posterior</a></li></ul></li>

<li><a href="Credible_interval" title="Credible interval">Credible interval</a></li>

<li><a href="Bayes_factor" title="Bayes factor">Bayes factor</a></li>

<li><a href="Bayes_estimator" title="Bayes estimator">Bayesian estimator</a>

<ul><li><a href="Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis654" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li>

<li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Confounding" title="Confounding">Confounding variable</a></li>

<li><a href="Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>

<li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>

<li><a href="Mixed_model" title="Mixed model">Mixed effects models</a></li>

<li><a href="Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li>

<li><a href="Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li>

<li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li>

<li><a href="General_linear_model" title="General linear model">General linear model</a></li>

<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>

<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>

<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>

<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>

<li><a href="Robust_regression" title="Robust regression">Robust</a></li>

<li><a href="Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Exponential_family" title="Exponential family">Exponential families</a></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic <span style="font-size: 85%;">(Bernoulli)</span></a> / <a href="Binomial_regression" title="Binomial regression">Binomial</a> / <a href="Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li>

<li><a href="Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li>

<li><a href="Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_multivariate_/_time-series_/_survival_analysis654" style="font-size:114%;margin:0 4em"><a href="Categorical_variable" title="Categorical variable">Categorical</a> / <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a> / <a href="Time_series" title="Time series">time-series</a> / <a href="Survival_analysis" title="Survival analysis">survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohen's_kappa" title="Cohen's kappa">Cohen's kappa</a></li>

<li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Graphical_model" title="Graphical model">Graphical model</a></li>

<li><a href="Poisson_regression" title="Poisson regression">Log-linear model</a></li>

<li><a href="McNemar's_test" title="McNemar's test">McNemar's test</a></li>

<li><a href="Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="General_linear_model" title="General linear model">Regression</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li>

<li><a href="Principal_component_analysis" title="Principal component analysis">Principal components</a></li>

<li><a href="Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li>

<li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li>

<li><a href="Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li>

<li><a href="Statistical_classification" title="Statistical classification">Classification</a></li>

<li><a href="Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a>

<ul><li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li>

<li><a href="Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a>

<ul><li><a href="Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a>

<ul><li><a href="Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li>

<li><a href="Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationarity</a></li>

<li><a href="Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li>

<li><a href="Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li>

<li><a href="Cointegration" title="Cointegration">Cointegration</a></li>

<li><a href="Structural_break" title="Structural break">Structural break</a></li>

<li><a href="Granger_causality" title="Granger causality">Granger causality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li>

<li><a href="Johansen_test" title="Johansen test">Johansen</a></li>

<li><a href="Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size: 85%;">(Ljung–Box)</span></a></li>

<li><a href="Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li>

<li><a href="Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a>

<ul><li><a href="Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li>

<li><a href="Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li>

<li><a href="Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size: 85%;">(Box–Jenkins)</span></a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li>

<li><a href="Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>

<li><a href="Wavelet" title="Wavelet">Wavelet</a></li>

<li><a href="Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li>

<li><a href="Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li>

<li><a href="Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li>

<li><a href="First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications654" style="font-size:114%;margin:0 4em"><a href="List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>

<li><a href="Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="Clinical_study_design" title="Clinical study design">studies</a></li>

<li><a href="Epidemiology" title="Epidemiology">Epidemiology</a></li>

<li><a href="Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chemometrics" title="Chemometrics">Chemometrics</a></li>

<li><a href="Methods_engineering" title="Methods engineering">Methods engineering</a></li>

<li><a href="Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li>

<li><a href="Statistical_process_control" title="Statistical process control">Process</a> / <a href="Quality_control" title="Quality control">quality control</a></li>

<li><a href="Reliability_engineering" title="Reliability engineering">Reliability</a></li>

<li><a href="System_identification" title="System identification">System identification</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_science" title="Actuarial science">Actuarial science</a></li>

<li><a href="Census" title="Census">Census</a></li>

<li><a href="Crime_statistics" title="Crime statistics">Crime statistics</a></li>

<li><a href="Demographic_statistics" title="Demographic statistics">Demography</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Jurimetrics" title="Jurimetrics">Jurimetrics</a></li>

<li><a href="National_accounts" title="National accounts">National accounts</a></li>

<li><a href="Official_statistics" title="Official statistics">Official statistics</a></li>

<li><a href="Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li>

<li><a href="Psychometrics" title="Psychometrics">Psychometrics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cartography" title="Cartography">Cartography</a></li>

<li><a href="Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li>

<li><a href="Geographic_information_system" title="Geographic information system">Geographic information system</a></li>

<li><a href="Geostatistics" title="Geostatistics">Geostatistics</a></li>

<li><a href="Kriging" title="Kriging">Kriging</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span><b>Category</b></li>

<li><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Statistics" class="extiw external" title="commons:Category:Statistics">Commons</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> <b>WikiProject</b></li></ul>

</div></td></tr></tbody></table></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q812535#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1301" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q812535#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1301" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Bayes-Entscheidungstheorie"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4144220-9">Germany</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Bayesian statistical decision theory"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85012506">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Bayesova teorie"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph135362&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007282424705171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/89628c4b-4062-4a41-9920-860f80a2eb33">Yale LUX</a></span></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-23" href="https://en.wikipedia.org/wiki/?title=Bayesian_inference&oldid=1302171891">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>