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Likelihood function

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

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

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

<ul><li><a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</a></li>

<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>

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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>

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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>

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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>

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<ul><li> <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">

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A likelihood function (often simply called the likelihood) measures how well a <a href="Statistical_model" title="Statistical model">statistical model</a> explains <a href="Realization_(probability)" title="Realization (probability)">observed data</a> by calculating the probability of seeing that data under different <a href="Statistical_parameter" title="Statistical parameter">parameter</a> values of the model. It is constructed from the <a href="Joint_probability_distribution" title="Joint probability distribution">joint probability distribution</a> of the <a href="Random_variable" title="Random variable">random variable</a> that (presumably) generated the observations.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a> When evaluated on the actual data points, it becomes a function solely of the model parameters.

In <a href="Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimation</a>, the <a href="Arg_max" title="Arg max">argument that maximizes</a> the likelihood function serves as a <a href="Point_estimation" title="Point estimation">point estimate</a> for the unknown parameter, while the <a href="Fisher_information" title="Fisher information">Fisher information</a> (often approximated by the likelihood's <a href="Hessian_matrix" title="Hessian matrix">Hessian matrix</a> at the maximum) gives an indication of the estimate's <a href="Precision_(statistics)" title="Precision (statistics)">precision</a>.

In contrast, in <a href="Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>, the estimate of interest is the converse of the likelihood, the so-called <a href="Posterior_probability" title="Posterior probability">posterior probability</a> of the parameter given the observed data, which is calculated via <a href="Bayes'_theorem" title="Bayes' theorem">Bayes' rule</a>.<a href="#cite_note-4">[4]</a>

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

The likelihood function, parameterized by a (possibly multivariate) parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

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

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</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy">, is usually defined differently for <a href="Continuous_or_discrete_variable" title="Continuous or discrete variable">discrete and continuous</a> <a href="Probability_distribution" title="Probability distribution">probability distributions</a> (a more general definition is discussed below). Given a probability density or mass function

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto f(x\mid \theta ),}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

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</mrow>

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

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</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy"> is a realization of the random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}">

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</mrow>

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

</semantics>

</math><img src="./8d80c41192705e1a6c6de1d65e16d7f70fbac391.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="{\textstyle X}" loading="lazy">, the likelihood function is

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<annotation encoding="application/x-tex">{\displaystyle \theta \mapsto f(x\mid \theta ),}</annotation>

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often written

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(\theta \mid x).}</annotation>

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</math>

In other words, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x\mid \theta )}">

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<annotation encoding="application/x-tex">{\textstyle f(x\mid \theta )}</annotation>

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</math><img src="./31ee488bd3e1b974f906f783752e2aaf94f468c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.445ex; height:2.843ex;" alt="{\textstyle f(x\mid \theta )}" loading="lazy"> is viewed as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

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</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

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</mrow>

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

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</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy"> fixed, it is a probability density function, and when viewed as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

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</mrow>

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

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</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy"> fixed, it is a likelihood function. In the <a href="Frequentist_probability" title="Frequentist probability">frequentist paradigm</a>, the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x\mid \theta )}">

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<annotation encoding="application/x-tex">{\textstyle f(x\mid \theta )}</annotation>

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</math><img src="./31ee488bd3e1b974f906f783752e2aaf94f468c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.445ex; height:2.843ex;" alt="{\textstyle f(x\mid \theta )}" loading="lazy"> is often avoided and instead <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x;\theta )}">

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</mrow>

<annotation encoding="application/x-tex">{\textstyle f(x;\theta )}</annotation>

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</math><img src="./d3ad5d00dd915a60af14332feaa1f6ed1c2b4c20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.542ex; height:2.843ex;" alt="{\textstyle f(x;\theta )}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x,\theta )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f(x,\theta )}</annotation>

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</math><img src="./ed2e60e029860ac0c7ad8d888da2b9c1b956ec92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.542ex; height:2.843ex;" alt="{\textstyle f(x,\theta )}" loading="lazy"> are used to indicate that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

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

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</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy"> is regarded as a fixed unknown quantity rather than as a <a href="Random_variable" title="Random variable">random variable</a> being conditioned on.

The likelihood function does not specify the probability that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

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

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</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy"> is the truth, given the observed sample <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X=x}">

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

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</math><img src="./1e3a35669a5ebcf051b61d6d7e49fbbea80a87b8.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="{\textstyle X=x}" loading="lazy">. Such an interpretation is a common error, with potentially disastrous consequences (see <a href="Prosecutor's_fallacy" class="mw-redirect" title="Prosecutor's fallacy">prosecutor's fallacy</a>).

<div class="mw-heading mw-heading3"><h3 id="Discrete_probability_distribution">Discrete probability distribution</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}">

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

</semantics>

</math><img src="./8d80c41192705e1a6c6de1d65e16d7f70fbac391.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="{\textstyle X}" loading="lazy"> be a discrete <a href="Random_variable" title="Random variable">random variable</a> with <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ad87bd7009e2a5c52bd0fb5a9bda9d8c1c23a79b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\textstyle p}" loading="lazy"> depending on a parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy">. Then the function

</mrow>

<msub>

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</msub>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(\theta \mid x)=p_{\theta }(x)=P_{\theta }(X=x),}</annotation>

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</math>

considered as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy">, is the likelihood function, given the <a href="Outcome_(probability)" title="Outcome (probability)">outcome</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy"> of the random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8d80c41192705e1a6c6de1d65e16d7f70fbac391.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="{\textstyle X}" loading="lazy">. Sometimes the probability of "the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8d80c41192705e1a6c6de1d65e16d7f70fbac391.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="{\textstyle X}" loading="lazy"> for the parameter value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy"> " is written as P(X = x | θ) or P(X = x; θ). The likelihood is the probability that a particular outcome <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy"> is observed when the true value of the parameter is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy">, equivalent to the probability mass on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy">; it is not a probability density over the parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy">. The likelihood, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {L}}(\theta \mid x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\mathcal {L}}(\theta \mid x)}</annotation>

</semantics>

</math><img src="./026d4a2d7fe2bb789c445f17aa5cfb9c4629be0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.77ex; height:2.843ex;" alt="{\textstyle {\mathcal {L}}(\theta \mid x)}" loading="lazy">, should not be confused with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P(\theta \mid x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle P(\theta \mid x)}</annotation>

</semantics>

</math><img src="./d1f68c0416700d8139a1488a22d1d0f4ca5c2f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\textstyle P(\theta \mid x)}" loading="lazy">, which is the posterior probability of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy"> given the data <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Example">Example</h4></div>

Consider a simple statistical model of a coin flip: a single parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p_{\text{H}}}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle p_{\text{H}}}</annotation>

</semantics>

</math><img src="./d4f0febe9d6852cbdaa077979a70b1e7882858c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.724ex; height:2.009ex;" alt="{\textstyle p_{\text{H}}}" loading="lazy"> that expresses the "fairness" of the coin. The parameter is the probability that a coin lands heads up ("H") when tossed. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p_{\text{H}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle p_{\text{H}}}</annotation>

</semantics>

</math><img src="./d4f0febe9d6852cbdaa077979a70b1e7882858c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.724ex; height:2.009ex;" alt="{\textstyle p_{\text{H}}}" loading="lazy"> can take on any value within the range 0.0 to 1.0. For a perfectly <a href="Fair_coin" title="Fair coin">fair coin</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p_{\text{H}}=0.5}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle p_{\text{H}}=0.5}</annotation>

</semantics>

</math><img src="./10517ec142215acbd645cdcb7dca982d345021fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.794ex; height:2.509ex;" alt="{\textstyle p_{\text{H}}=0.5}" loading="lazy">.

Imagine flipping a fair coin twice, and observing two heads in two tosses ("HH"). Assuming that each successive coin flip is <a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">i.i.d.</a>, then the probability of observing HH is

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle P({\text{HH}}\mid p_{\text{H}}=0.5)=0.5^{2}=0.25.}</annotation>

</semantics>

</math>

Equivalently, the likelihood of observing "HH" assuming <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p_{\text{H}}=0.5}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle p_{\text{H}}=0.5}</annotation>

</semantics>

</math><img src="./10517ec142215acbd645cdcb7dca982d345021fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.794ex; height:2.509ex;" alt="{\textstyle p_{\text{H}}=0.5}" loading="lazy"> is

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(p_{\text{H}}=0.5\mid {\text{HH}})=0.25.}</annotation>

</semantics>

</math>

This is not the same as saying that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P(p_{\text{H}}=0.5\mid HH)=0.25}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle P(p_{\text{H}}=0.5\mid HH)=0.25}</annotation>

</semantics>

</math><img src="./dea12069caf3101ef355008cb5e37291314126fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.556ex; height:2.843ex;" alt="{\textstyle P(p_{\text{H}}=0.5\mid HH)=0.25}" loading="lazy">, a conclusion which could only be reached via <a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a> given knowledge about the marginal probabilities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P(p_{\text{H}}=0.5)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle P(p_{\text{H}}=0.5)}</annotation>

</semantics>

</math><img src="./5cd2a012dd72827caadc76af9e62ce5f3bccfaf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.259ex; height:2.843ex;" alt="{\textstyle P(p_{\text{H}}=0.5)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P({\text{HH}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle P({\text{HH}})}</annotation>

</semantics>

</math><img src="./63352ffda21c7e8fce881e6f23c832fd88cdc774.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.041ex; height:2.843ex;" alt="{\textstyle P({\text{HH}})}" loading="lazy">.

Now suppose that the coin is not a fair coin, but instead that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p_{\text{H}}=0.3}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle p_{\text{H}}=0.3}</annotation>

</semantics>

</math><img src="./e21e5f0e22ba48b6c05e484e67178e646f1b0e98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.794ex; height:2.509ex;" alt="{\textstyle p_{\text{H}}=0.3}" loading="lazy">. Then the probability of two heads on two flips is

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P({\text{HH}}\mid p_{\text{H}}=0.3)=0.3^{2}=0.09.}</annotation>

</semantics>

</math>

Hence

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(p_{\text{H}}=0.3\mid {\text{HH}})=0.09.}</annotation>

</semantics>

</math>

More generally, for each value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p_{\text{H}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle p_{\text{H}}}</annotation>

</semantics>

</math><img src="./d4f0febe9d6852cbdaa077979a70b1e7882858c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.724ex; height:2.009ex;" alt="{\textstyle p_{\text{H}}}" loading="lazy">, we can calculate the corresponding likelihood. The result of such calculations is displayed in Figure 1. The integral of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {L}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\mathcal {L}}}</annotation>

</semantics>

</math><img src="./9e97aae95c92641d7ecba829fa92d6fe23888465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\textstyle {\mathcal {L}}}" loading="lazy"> over [0, 1] is 1/3; likelihoods need not integrate or sum to one over the parameter space.

<div class="mw-heading mw-heading3"><h3 id="Continuous_probability_distribution">Continuous probability distribution</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8d80c41192705e1a6c6de1d65e16d7f70fbac391.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="{\textstyle X}" loading="lazy"> be a <a href="Random_variable" title="Random variable">random variable</a> following an <a href="Probability_distribution#Continuous_probability_distribution" title="Probability distribution">absolutely continuous probability distribution</a> with <a href="Probability_density_function" title="Probability density function">density function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e1b77076edca76caf3331d0551d1645b8f678283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}" loading="lazy"> (a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy">) which depends on a parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy">. Then the function

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(\theta \mid x)=f_{\theta }(x),}</annotation>

</semantics>

</math>

considered as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy">, is the likelihood function (of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy">, given the <a href="Outcome_(probability)" title="Outcome (probability)">outcome</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X=x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1e3a35669a5ebcf051b61d6d7e49fbbea80a87b8.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="{\textstyle X=x}" loading="lazy">). Again, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {L}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\mathcal {L}}}</annotation>

</semantics>

</math><img src="./9e97aae95c92641d7ecba829fa92d6fe23888465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\textstyle {\mathcal {L}}}" loading="lazy"> is not a probability density or mass function over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy">, despite being a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy"> given the observation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X=x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1e3a35669a5ebcf051b61d6d7e49fbbea80a87b8.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="{\textstyle X=x}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Relationship_between_the_likelihood_and_probability_density_functions">Relationship between the likelihood and probability density functions</h4></div>

The use of the <a href="Probability_density_function" title="Probability density function">probability density</a> in specifying the likelihood function above is justified as follows. Given an observation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle x_{j}}</annotation>

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</math><img src="./bed4bfb1ba3d13293c3ea916b928dfe07532195f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\textstyle x_{j}}" loading="lazy">, the likelihood for the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [x_{j},x_{j}+h]}">

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\textstyle [x_{j},x_{j}+h]}</annotation>

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</math><img src="./84900da400b7056cb56e8531e4d86d58d82d584b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.986ex; height:3.009ex;" alt="{\textstyle [x_{j},x_{j}+h]}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle h>0}">

</mstyle>

</mrow>

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

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</math><img src="./1d0e64989e5d2d0a6bbb414bd390c0dadb2bc514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\textstyle h>0}" loading="lazy"> is a constant, is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])}">

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<msub>

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<annotation encoding="application/x-tex">{\textstyle {\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])}</annotation>

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</math><img src="./5b9fa9e5e896f5bba0aba4bcc54d301dc14017db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.597ex; height:3.009ex;" alt="{\textstyle {\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])}" loading="lazy">. Observe that

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}{\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h]),}</annotation>

</semantics>

</math>

since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle h}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./13fda070627ca694f85f588a432f8158cc4df1e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\textstyle h}" loading="lazy"> is positive and constant. Because

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</munder>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}{\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}\Pr(x_{j}\leq x\leq x_{j}+h\mid \theta )=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x\mid \theta )\,dx,}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x\mid \theta )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f(x\mid \theta )}</annotation>

</semantics>

</math><img src="./31ee488bd3e1b974f906f783752e2aaf94f468c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.445ex; height:2.843ex;" alt="{\textstyle f(x\mid \theta )}" loading="lazy"> is the probability density function, it follows that

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<mfrac>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x\mid \theta )\,dx.}</annotation>

</semantics>

</math>

The first <a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a> provides that

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{h\to 0^{+}}{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x\mid \theta )\,dx=f(x_{j}\mid \theta ).}</annotation>

</semantics>

</math>

Then

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</munder>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x_{j})&=\mathop {\operatorname {arg\,max} } _{\theta }\left[\lim _{h\to 0^{+}}{\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])\right]\\[4pt]&=\mathop {\operatorname {arg\,max} } _{\theta }\left[\lim _{h\to 0^{+}}{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x\mid \theta )\,dx\right]\\[4pt]&=\mathop {\operatorname {arg\,max} } _{\theta }f(x_{j}\mid \theta ).\end{aligned}}}</annotation>

</semantics>

</math>

Therefore,

</mrow>

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</mrow>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x_{j})=\mathop {\operatorname {arg\,max} } _{\theta }f(x_{j}\mid \theta ),}</annotation>

</semantics>

</math>

and so maximizing the probability density at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x_{j}}</annotation>

</semantics>

</math><img src="./bed4bfb1ba3d13293c3ea916b928dfe07532195f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\textstyle x_{j}}" loading="lazy"> amounts to maximizing the likelihood of the specific observation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x_{j}}</annotation>

</semantics>

</math><img src="./bed4bfb1ba3d13293c3ea916b928dfe07532195f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\textstyle x_{j}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="In_general">In general</h3></div>

In <a href="Probability_theory#Measure-theoretic_probability_theory" title="Probability theory">measure-theoretic probability theory</a>, the <a href="Probability_density_function" title="Probability density function">density function</a> is defined as the <a href="Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">Radon–Nikodym derivative</a> of the probability distribution relative to a common dominating measure.<a href="#cite_note-5">[5]</a> The likelihood function is this density interpreted as a function of the parameter, rather than the random variable.<a href="#cite_note-Shao03-6">[6]</a> Thus, we can construct a likelihood function for any distribution, whether discrete, continuous, a mixture, or otherwise. (Likelihoods are comparable, e.g. for parameter estimation, only if they are Radon–Nikodym derivatives with respect to the same dominating measure.)

The above discussion of the likelihood for discrete random variables uses the <a href="Counting_measure" title="Counting measure">counting measure</a>, under which the probability density at any outcome equals the probability of that outcome.

<div class="mw-heading mw-heading3"><h3 id="Likelihoods_for_mixed_continuous–discrete_distributions">Likelihoods for mixed continuous–discrete distributions</h3></div>

The above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components. Suppose that the distribution consists of a number of discrete probability masses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p_{k}(\theta )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle p_{k}(\theta )}</annotation>

</semantics>

</math><img src="./6057d0999196f6eab9a0b5a130435aa7d4a34d56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.247ex; height:2.843ex;" alt="{\textstyle p_{k}(\theta )}" loading="lazy"> and a density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x\mid \theta )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f(x\mid \theta )}</annotation>

</semantics>

</math><img src="./31ee488bd3e1b974f906f783752e2aaf94f468c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.445ex; height:2.843ex;" alt="{\textstyle f(x\mid \theta )}" loading="lazy">, where the sum of all the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ad87bd7009e2a5c52bd0fb5a9bda9d8c1c23a79b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\textstyle p}" loading="lazy">'s added to the integral of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e1b77076edca76caf3331d0551d1645b8f678283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}" loading="lazy"> is always one. Assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to the density component, the likelihood function for an observation from the continuous component can be dealt with in the manner shown above. For an observation from the discrete component, the likelihood function for an observation from the discrete component is simply

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(\theta \mid x)=p_{k}(\theta ),}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0d5595fc0c47452f8fc2aa6e29c3611f047714b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}" loading="lazy"> is the index of the discrete probability mass corresponding to observation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy">, because maximizing the probability mass (or probability) at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy"> amounts to maximizing the likelihood of the specific observation.

The fact that the likelihood function can be defined in a way that includes contributions that are not commensurate (the density and the probability mass) arises from the way in which the likelihood function is defined up to a constant of proportionality, where this "constant" can change with the observation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy">, but not with the parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Regularity_conditions">Regularity conditions</h3></div>

In the context of parameter estimation, the likelihood function is usually assumed to obey certain conditions, known as regularity conditions. These conditions are assumed in various proofs involving likelihood functions, and need to be verified in each particular application. For maximum likelihood estimation, the existence of a global maximum of the likelihood function is of the utmost importance. By the <a href="Extreme_value_theorem" title="Extreme value theorem">extreme value theorem</a>, it suffices that the likelihood function is <a href="Continuous_function" title="Continuous function">continuous</a> on a <a href="Compactness" class="mw-redirect" title="Compactness">compact</a> parameter space for the maximum likelihood estimator to exist.<a href="#cite_note-7">[7]</a> While the continuity assumption is usually met, the compactness assumption about the parameter space is often not, as the bounds of the true parameter values might be unknown. In that case, <a href="Concave_function" title="Concave function">concavity</a> of the likelihood function plays a key role.

More specifically, if the likelihood function is twice continuously differentiable on the <var>k</var>-dimensional parameter space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4ce78034dc51a51642f247b9746264abd71704e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\textstyle \Theta }" loading="lazy"> assumed to be an <a href="Open_set" title="Open set">open</a> <a href="Connected_space" title="Connected space">connected</a> subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbb {R} ^{k}\,,}">

<msup>

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</mrow>

</msup>

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<annotation encoding="application/x-tex">{\textstyle \mathbb {R} ^{k}\,,}</annotation>

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</math><img src="./7da3f2eb505e224fa42ea63d6c2d5eab86e9607a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.801ex; height:3.009ex;" alt="{\textstyle \mathbb {R} ^{k}\,,}" loading="lazy"> there exists a unique maximum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\hat {\theta }}\in \Theta }">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\hat {\theta }}\in \Theta }</annotation>

</semantics>

</math><img src="./6b4409d0f523177b6608c9acd7ef500015724c87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.005ex; height:2.843ex;" alt="{\textstyle {\hat {\theta }}\in \Theta }" loading="lazy"> if the <a href="Hessian_matrix" title="Hessian matrix">matrix of second partials</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} (\theta )\equiv \left[\,{\frac {\partial ^{2}L}{\,\partial \theta _{i}\,\partial \theta _{j}\,}}\,\right]_{i,j=1,1}^{n_{\mathrm {i} },n_{\mathrm {j} }}\;}</annotation>

</semantics>

</math> is <a href="Negative_definite" class="mw-redirect" title="Negative definite">negative definite</a> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \,\theta \in \Theta \,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \,\theta \in \Theta \,}</annotation>

</semantics>

</math><img src="./c93031bf7267da2f4b00a267a758ef75e4a761cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.513ex; height:2.176ex;" alt="{\textstyle \,\theta \in \Theta \,}" loading="lazy"> at which the gradient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \;\nabla L\equiv \left[\,{\frac {\partial L}{\,\partial \theta _{i}\,}}\,\right]_{i=1}^{n_{\mathrm {i} }}\;}">

<mrow>

<mrow>

<mfrac>

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<mrow>

<msub>

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<annotation encoding="application/x-tex">{\textstyle \;\nabla L\equiv \left[\,{\frac {\partial L}{\,\partial \theta _{i}\,}}\,\right]_{i=1}^{n_{\mathrm {i} }}\;}</annotation>

</semantics>

</math><img src="./6608634b066833afa567e2de871fb2497bcc2587.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.715ex; height:4.843ex;" alt="{\textstyle \;\nabla L\equiv \left[\,{\frac {\partial L}{\,\partial \theta _{i}\,}}\,\right]_{i=1}^{n_{\mathrm {i} }}\;}" loading="lazy"> vanishes,

and if the likelihood function approaches a constant on the <a href="Boundary_(topology)" title="Boundary (topology)">boundary</a> of the parameter space, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \;\partial \Theta \;,}">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle \;\partial \Theta \;,}</annotation>

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</math><img src="./3ba18347a17cf4b3cea67cd4700dbf8f2dcfd0d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.063ex; height:2.509ex;" alt="{\textstyle \;\partial \Theta \;,}" loading="lazy"> i.e.,

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<annotation encoding="application/x-tex">{\displaystyle \lim _{\theta \to \partial \Theta }L(\theta )=0\;,}</annotation>

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which may include the points at infinity if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \,\Theta \,}">

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<annotation encoding="application/x-tex">{\textstyle \,\Theta \,}</annotation>

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</math><img src="./91c8a7232e57689ecf9ff2b326c081e8c9887e39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.582ex; height:2.176ex;" alt="{\textstyle \,\Theta \,}" loading="lazy"> is unbounded. Mäkeläinen and co-authors prove this result using <a href="Morse_theory" title="Morse theory">Morse theory</a> while informally appealing to a mountain pass property.<a href="#cite_note-8">[8]</a> Mascarenhas restates their proof using the <a href="Mountain_pass_theorem" title="Mountain pass theorem">mountain pass theorem</a>.<a href="#cite_note-9">[9]</a>

In the proofs of <a href="Consistent_estimator" title="Consistent estimator">consistency</a> and asymptotic normality of the maximum likelihood estimator, additional assumptions are made about the probability densities that form the basis of a particular likelihood function. These conditions were first established by Chanda.<a href="#cite_note-10">[10]</a> In particular, for <a href="Almost_all" title="Almost all">almost all</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

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</math><img src="./aa994738de0d896807b240978e6ae01df896c728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.16ex; height:2.509ex;" alt="{\textstyle \,\theta \in \Theta \,,}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \log f}{\partial \theta _{r}}}\,,\quad {\frac {\partial ^{2}\log f}{\partial \theta _{r}\partial \theta _{s}}}\,,\quad {\frac {\partial ^{3}\log f}{\partial \theta _{r}\,\partial \theta _{s}\,\partial \theta _{t}}}\,}</annotation>

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exist for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \,r,s,t=1,2,\ldots ,k\,}">

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<annotation encoding="application/x-tex">{\textstyle \,r,s,t=1,2,\ldots ,k\,}</annotation>

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</math><img src="./5d2026dbba273fedfd5390626262d20dad5d0d9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.668ex; height:2.509ex;" alt="{\textstyle \,r,s,t=1,2,\ldots ,k\,}" loading="lazy"> in order to ensure the existence of a <a href="Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a>. Second, for almost all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

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</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy"> and for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \,\theta \in \Theta \,}">

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<annotation encoding="application/x-tex">{\textstyle \,\theta \in \Theta \,}</annotation>

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</math><img src="./c93031bf7267da2f4b00a267a758ef75e4a761cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.513ex; height:2.176ex;" alt="{\textstyle \,\theta \in \Theta \,}" loading="lazy"> it must be that

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<annotation encoding="application/x-tex">{\displaystyle \left|{\frac {\partial f}{\partial \theta _{r}}}\right|<F_{r}(x)\,,\quad \left|{\frac {\partial ^{2}f}{\partial \theta _{r}\,\partial \theta _{s}}}\right|<F_{rs}(x)\,,\quad \left|{\frac {\partial ^{3}f}{\partial \theta _{r}\,\partial \theta _{s}\,\partial \theta _{t}}}\right|<H_{rst}(x)}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H}">

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</math><img src="./2435ce1b360a5f6f5a0ab5acbb3c672bf229dac7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\textstyle H}" loading="lazy"> is such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \,\int _{-\infty }^{\infty }H_{rst}(z)\mathrm {d} z\leq M<\infty \;.}">

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<annotation encoding="application/x-tex">{\textstyle \,\int _{-\infty }^{\infty }H_{rst}(z)\mathrm {d} z\leq M<\infty \;.}</annotation>

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</math><img src="./00a72026755a407ee47ad3a3dc252f55436c4f6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.828ex; height:3.343ex;" alt="{\textstyle \,\int _{-\infty }^{\infty }H_{rst}(z)\mathrm {d} z\leq M<\infty \;.}" loading="lazy"> This boundedness of the derivatives is needed to allow for <a href="Differentiation_under_the_integral_sign" class="mw-redirect" title="Differentiation under the integral sign">differentiation under the integral sign</a>. And lastly, it is assumed that the <a href="Information_matrix" class="mw-redirect" title="Information matrix">information matrix</a>,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {I} (\theta )=\int _{-\infty }^{\infty }{\frac {\partial \log f}{\partial \theta _{r}}}\ {\frac {\partial \log f}{\partial \theta _{s}}}\ f\ \mathrm {d} z}</annotation>

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is <a href="Positive_definite" class="mw-redirect" title="Positive definite">positive definite</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \,\left|\mathbf {I} (\theta )\right|\,}">

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<annotation encoding="application/x-tex">{\textstyle \,\left|\mathbf {I} (\theta )\right|\,}</annotation>

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</math><img src="./6d0645841af58ef06f7b71e46a0dcffdcf4c9e0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.981ex; height:2.843ex;" alt="{\textstyle \,\left|\mathbf {I} (\theta )\right|\,}" loading="lazy"> is finite. This ensures that the <a href="Score_(statistics)" class="mw-redirect" title="Score (statistics)">score</a> has a finite variance.<a href="#cite_note-11">[11]</a>

The above conditions are sufficient, but not necessary. That is, a model that does not meet these regularity conditions may or may not have a maximum likelihood estimator of the properties mentioned above. Further, in case of non-independently or non-identically distributed observations additional properties may need to be assumed.

In Bayesian statistics, almost identical regularity conditions are imposed on the likelihood function in order to proof asymptotic normality of the <a href="Posterior_probability" title="Posterior probability">posterior probability</a>,<a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a> and therefore to justify a <a href="Laplace_approximation" class="mw-redirect" title="Laplace approximation">Laplace approximation</a> of the posterior in large samples.<a href="#cite_note-14">[14]</a>

<div class="mw-heading mw-heading2"><h2 id="Likelihood_ratio_and_relative_likelihood">Likelihood ratio and relative likelihood</h2></div>

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

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Pseudo-R-squared" title="Pseudo-R-squared">Pseudo-R-squared</a></div>

<div class="mw-heading mw-heading3"><h3 id="Likelihood_ratio">Likelihood ratio</h3></div>

<div role="note" class="hatnote navigation-not-searchable">This section is about the likelihood ratio in general. For the use of likelihood ratios in interpreting diagnostic tests, see <a href="Likelihood_ratios_in_diagnostic_testing" title="Likelihood ratios in diagnostic testing">Likelihood ratios in diagnostic testing</a>. For the statistical test to compare goodness of fit, see <a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a>.</div>

A likelihood ratio is the ratio of any two specified likelihoods, frequently written as:

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (\theta _{1}:\theta _{2}\mid x)={\frac {{\mathcal {L}}(\theta _{1}\mid x)}{{\mathcal {L}}(\theta _{2}\mid x)}}.}</annotation>

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The likelihood ratio is central to <a href="Likelihoodist_statistics" title="Likelihoodist statistics">likelihoodist statistics</a>: the <a href="Law_of_likelihood" class="mw-redirect" title="Law of likelihood">law of likelihood</a> states that the degree to which data (considered as evidence) supports one parameter value versus another is measured by the likelihood ratio.

In <a href="Frequentist_inference" title="Frequentist inference">frequentist inference</a>, the likelihood ratio is the basis for a <a href="Test_statistic" title="Test statistic">test statistic</a>, the so-called <a href="Likelihood-ratio_test" title="Likelihood-ratio test">likelihood-ratio test</a>. By the <a href="Neyman%E2%80%93Pearson_lemma" title="Neyman–Pearson lemma">Neyman–Pearson lemma</a>, this is the most <a href="Statistical_power" class="mw-redirect" title="Statistical power">powerful</a> test for comparing two <a href="Simple_hypothesis" class="mw-redirect" title="Simple hypothesis">simple hypotheses</a> at a given <a href="Significance_level" class="mw-redirect" title="Significance level">significance level</a>. Numerous other tests can be viewed as likelihood-ratio tests or approximations thereof.<a href="#cite_note-15">[15]</a> The asymptotic distribution of the log-likelihood ratio, considered as a test statistic, is given by <a href="Wilks'_theorem" title="Wilks' theorem">Wilks' theorem</a>.

The likelihood ratio is also of central importance in <a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</a>, where it is known as the <a href="Bayes_factor" title="Bayes factor">Bayes factor</a>, and is used in <a href="Bayes'_rule" class="mw-redirect" title="Bayes' rule">Bayes' rule</a>. Stated in terms of <a href="Odds" title="Odds">odds</a>, Bayes' rule states that the posterior odds of two alternatives, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}">

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</math><img src="./3ec73b8bc9abc3efb934f5a6ec2803713771f4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{2}}" loading="lazy">⁠, given an event ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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The likelihood ratio is not directly used in AIC-based statistics. Instead, what is used is the relative likelihood of models (see below).

In <a href="Evidence-based_medicine" title="Evidence-based medicine">evidence-based medicine</a>, likelihood ratios <a href="Likelihood_ratios_in_diagnostic_testing" title="Likelihood ratios in diagnostic testing">are used in diagnostic testing</a> to assess the value of performing a <a href="Diagnostic_test" class="mw-redirect" title="Diagnostic test">diagnostic test</a>.

<div class="mw-heading mw-heading3"><h3 id="Relative_likelihood_function">Relative likelihood function</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Relative_likelihood" title="Relative likelihood">Relative likelihood</a></div>

Since the actual value of the likelihood function depends on the sample, it is often convenient to work with a standardized measure. Suppose that the <a href="Maximum_likelihood_estimate" class="mw-redirect" title="Maximum likelihood estimate">maximum likelihood estimate</a> for the parameter θ is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\hat {\theta }}}">

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</math><img src="./12604986cd8d01bc09f476d5baafbfbcae2367af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\textstyle {\hat {\theta }}}" loading="lazy">. The relative likelihood of θ is defined to be<a href="#cite_note-Kalbfleisch-16">[16]</a><a href="#cite_note-17">[17]</a><a href="#cite_note-Sprott-18">[18]</a><a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a>

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Thus, the relative likelihood is the likelihood ratio (discussed above) with the fixed denominator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {L}}({\hat {\theta }})}">

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<div class="mw-heading mw-heading4"><h4 id="Likelihood_region">Likelihood region</h4></div>

A likelihood region is the set of all values of θ whose relative likelihood is greater than or equal to a given threshold. In terms of percentages, a p% likelihood region for θ is defined to be<a href="#cite_note-Kalbfleisch-16">[16]</a><a href="#cite_note-Sprott-18">[18]</a><a href="#cite_note-Rossi2018-21">[21]</a>

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If θ is a single real parameter, a p% likelihood region will usually comprise an <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> of real values. If the region does comprise an interval, then it is called a likelihood interval.<a href="#cite_note-Kalbfleisch-16">[16]</a><a href="#cite_note-Sprott-18">[18]</a><a href="#cite_note-Hudson-22">[22]</a>

Likelihood intervals, and more generally likelihood regions, are used for <a href="Interval_estimation" title="Interval estimation">interval estimation</a> within likelihoodist statistics: they are similar to <a href="Confidence_interval" title="Confidence interval">confidence intervals</a> in frequentist statistics and <a href="Credible_interval" title="Credible interval">credible intervals</a> in Bayesian statistics. Likelihood intervals are interpreted directly in terms of relative likelihood, not in terms of <a href="Coverage_probability" title="Coverage probability">coverage probability</a> (frequentism) or <a href="Posterior_probability" title="Posterior probability">posterior probability</a> (Bayesianism).

Given a model, likelihood intervals can be compared to confidence intervals. If θ is a single real parameter, then under certain conditions, a 14.65% likelihood interval (about 1:7 likelihood) for θ will be the same as a 95% confidence interval (19/20 coverage probability).<a href="#cite_note-Kalbfleisch-16">[16]</a><a href="#cite_note-Rossi2018-21">[21]</a> In a slightly different formulation suited to the use of log-likelihoods (see <a href="Likelihood-ratio_test#Distribution:_Wilks.27_theorem" title="Likelihood-ratio test">Wilks' theorem</a>), the test statistic is twice the difference in log-likelihoods and the probability distribution of the test statistic is approximately a <a href="Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> with degrees-of-freedom (df) equal to the difference in df's between the two models (therefore, the e−2 likelihood interval is the same as the 0.954 confidence interval; assuming difference in df's to be 1).<a href="#cite_note-Rossi2018-21">[21]</a><a href="#cite_note-Hudson-22">[22]</a>

<div class="mw-heading mw-heading2"><h2 id="Likelihoods_that_eliminate_nuisance_parameters">Likelihoods that eliminate nuisance parameters</h2></div>

In many cases, the likelihood is a function of more than one parameter but interest focuses on the estimation of only one, or at most a few of them, with the others being considered as <a href="Nuisance_parameter" title="Nuisance parameter">nuisance parameters</a>. Several alternative approaches have been developed to eliminate such nuisance parameters, so that a likelihood can be written as a function of only the parameter (or parameters) of interest: the main approaches are profile, conditional, and marginal likelihoods.<a href="#cite_note-23">[23]</a><a href="#cite_note-24">[24]</a> These approaches are also useful when a high-dimensional likelihood surface needs to be reduced to one or two parameters of interest in order to allow a <a href="Graph_of_a_function" title="Graph of a function">graph</a>.

<div class="mw-heading mw-heading3"><h3 id="Profile_likelihood">Profile likelihood</h3></div>

It is possible to reduce the dimensions by concentrating the likelihood function for a subset of parameters by expressing the nuisance parameters as functions of the parameters of interest and replacing them in the likelihood function.<a href="#cite_note-25">[25]</a><a href="#cite_note-26">[26]</a> In general, for a likelihood function depending on the parameter vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {\theta } }">

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</math><img src="./cef53d598a812dba39d8ff5c8b57484a80e682db.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="{\textstyle \mathbf {\theta } }" loading="lazy"> that can be partitioned into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {\theta } =\left(\mathbf {\theta } _{1}:\mathbf {\theta } _{2}\right)}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {\theta } =\left(\mathbf {\theta } _{1}:\mathbf {\theta } _{2}\right)}</annotation>

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</math><img src="./dcbbf1d26eb2897a7517953f1071f51097fcd293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.225ex; height:2.843ex;" alt="{\textstyle \mathbf {\theta } =\left(\mathbf {\theta } _{1}:\mathbf {\theta } _{2}\right)}" loading="lazy">, and where a correspondence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {\hat {\theta }} _{2}=\mathbf {\hat {\theta }} _{2}\left(\mathbf {\theta } _{1}\right)}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {\hat {\theta }} _{2}=\mathbf {\hat {\theta }} _{2}\left(\mathbf {\theta } _{1}\right)}</annotation>

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</math><img src="./57d71c203595bbf8ba0ca64d467d9d2ed16e4258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.608ex; height:3.343ex;" alt="{\textstyle \mathbf {\hat {\theta }} _{2}=\mathbf {\hat {\theta }} _{2}\left(\mathbf {\theta } _{1}\right)}" loading="lazy"> can be determined explicitly, concentration reduces <a href="Computational_complexity" title="Computational complexity">computational burden</a> of the original maximization problem.<a href="#cite_note-27">[27]</a>

For instance, in a <a href="Linear_regression" title="Linear regression">linear regression</a> with normally distributed errors, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {y} =\mathbf {X} \beta +u}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {y} =\mathbf {X} \beta +u}</annotation>

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</math><img src="./80bcf4b2b72273e6bcee572fca483134c7b6a216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.031ex; height:2.509ex;" alt="{\textstyle \mathbf {y} =\mathbf {X} \beta +u}" loading="lazy">, the coefficient vector could be <a href="Partition_of_a_set" title="Partition of a set">partitioned</a> into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta =\left[\beta _{1}:\beta _{2}\right]}">

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<annotation encoding="application/x-tex">{\textstyle \beta =\left[\beta _{1}:\beta _{2}\right]}</annotation>

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</math><img src="./6bb24b21da678e7a894d6195d63aaed0451bd1db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.401ex; height:2.843ex;" alt="{\textstyle \beta =\left[\beta _{1}:\beta _{2}\right]}" loading="lazy"> (and consequently the <a href="Design_matrix" title="Design matrix">design matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {X} =\left[\mathbf {X} _{1}:\mathbf {X} _{2}\right]}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {X} =\left[\mathbf {X} _{1}:\mathbf {X} _{2}\right]}</annotation>

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</math><img src="./7e735d0d998420deb9b647276ddf823116e4d7c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.496ex; height:2.843ex;" alt="{\textstyle \mathbf {X} =\left[\mathbf {X} _{1}:\mathbf {X} _{2}\right]}" loading="lazy">). Maximizing with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta _{2}}">

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<annotation encoding="application/x-tex">{\textstyle \beta _{2}}</annotation>

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</math><img src="./b92793c9c5bf00c01ba181622d505da61ce78ac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\textstyle \beta _{2}}" loading="lazy"> yields an optimal value function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta _{2}(\beta _{1})=\left(\mathbf {X} _{2}^{\mathsf {T}}\mathbf {X} _{2}\right)^{-1}\mathbf {X} _{2}^{\mathsf {T}}\left(\mathbf {y} -\mathbf {X} _{1}\beta _{1}\right)}">

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<annotation encoding="application/x-tex">{\textstyle \beta _{2}(\beta _{1})=\left(\mathbf {X} _{2}^{\mathsf {T}}\mathbf {X} _{2}\right)^{-1}\mathbf {X} _{2}^{\mathsf {T}}\left(\mathbf {y} -\mathbf {X} _{1}\beta _{1}\right)}</annotation>

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</math><img src="./c76ee8f29b5f9ec4c0ee889f06e2822c934a8a53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.818ex; height:3.843ex;" alt="{\textstyle \beta _{2}(\beta _{1})=\left(\mathbf {X} _{2}^{\mathsf {T}}\mathbf {X} _{2}\right)^{-1}\mathbf {X} _{2}^{\mathsf {T}}\left(\mathbf {y} -\mathbf {X} _{1}\beta _{1}\right)}" loading="lazy">. Using this result, the maximum likelihood estimator for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta _{1}}">

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<annotation encoding="application/x-tex">{\textstyle \beta _{1}}</annotation>

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</math><img src="./20c28f9a17352245b8e1bedd79cfe60e4418b2dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\textstyle \beta _{1}}" loading="lazy"> can then be derived as

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}_{1}=\left(\mathbf {X} _{1}^{\mathsf {T}}\left(\mathbf {I} -\mathbf {P} _{2}\right)\mathbf {X} _{1}\right)^{-1}\mathbf {X} _{1}^{\mathsf {T}}\left(\mathbf {I} -\mathbf {P} _{2}\right)\mathbf {y} }</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {P} _{2}=\mathbf {X} _{2}\left(\mathbf {X} _{2}^{\mathsf {T}}\mathbf {X} _{2}\right)^{-1}\mathbf {X} _{2}^{\mathsf {T}}}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {P} _{2}=\mathbf {X} _{2}\left(\mathbf {X} _{2}^{\mathsf {T}}\mathbf {X} _{2}\right)^{-1}\mathbf {X} _{2}^{\mathsf {T}}}</annotation>

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</math><img src="./6f993f51de99cfb3eea9501dfd0572309c3752a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.331ex; height:3.843ex;" alt="{\textstyle \mathbf {P} _{2}=\mathbf {X} _{2}\left(\mathbf {X} _{2}^{\mathsf {T}}\mathbf {X} _{2}\right)^{-1}\mathbf {X} _{2}^{\mathsf {T}}}" loading="lazy"> is the <a href="Projection_matrix" title="Projection matrix">projection matrix</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {X} _{2}}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {X} _{2}}</annotation>

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</math><img src="./3fd2e7174444b086a8d93d5e6c562d111475aec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.074ex; height:2.509ex;" alt="{\textstyle \mathbf {X} _{2}}" loading="lazy">. This result is known as the <a href="Frisch%E2%80%93Waugh%E2%80%93Lovell_theorem" title="Frisch–Waugh–Lovell theorem">Frisch–Waugh–Lovell theorem</a>.

Since graphically the procedure of concentration is equivalent to slicing the likelihood surface along the ridge of values of the nuisance parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta _{2}}">

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</math><img src="./b92793c9c5bf00c01ba181622d505da61ce78ac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\textstyle \beta _{2}}" loading="lazy"> that maximizes the likelihood function, creating an <a href="Contour_line" title="Contour line">isometric</a> <a href="Topographic_profile" title="Topographic profile">profile</a> of the likelihood function for a given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta _{1}}">

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</math><img src="./20c28f9a17352245b8e1bedd79cfe60e4418b2dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\textstyle \beta _{1}}" loading="lazy">, the result of this procedure is also known as profile likelihood.<a href="#cite_note-28">[28]</a><a href="#cite_note-29">[29]</a> In addition to being graphed, the profile likelihood can also be used to compute <a href="Confidence_interval" title="Confidence interval">confidence intervals</a> that often have better small-sample properties than those based on asymptotic <a href="Standard_error_(statistics)" class="mw-redirect" title="Standard error (statistics)">standard errors</a> calculated from the full likelihood.<a href="#cite_note-30">[30]</a><a href="#cite_note-31">[31]</a>

<div class="mw-heading mw-heading3"><h3 id="Conditional_likelihood">Conditional likelihood</h3></div>

Sometimes it is possible to find a <a href="Sufficient_statistic" title="Sufficient statistic">sufficient statistic</a> for the nuisance parameters, and conditioning on this statistic results in a likelihood which does not depend on the nuisance parameters.<a href="#cite_note-32">[32]</a>

One example occurs in 2×2 tables, where conditioning on all four marginal totals leads to a conditional likelihood based on the non-central <a href="Hypergeometric_distribution" title="Hypergeometric distribution">hypergeometric distribution</a>. This form of conditioning is also the basis for <a href="Fisher's_exact_test" title="Fisher's exact test">Fisher's exact test</a>.

<div class="mw-heading mw-heading3"><h3 id="Marginal_likelihood">Marginal likelihood</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Marginal_likelihood" title="Marginal likelihood">Marginal likelihood</a></div>

Sometimes we can remove the nuisance parameters by considering a likelihood based on only part of the information in the data, for example by using the set of ranks rather than the numerical values. Another example occurs in linear <a href="Mixed_model" title="Mixed model">mixed models</a>, where considering a likelihood for the residuals only after fitting the fixed effects leads to <a href="Residual_maximum_likelihood" class="mw-redirect" title="Residual maximum likelihood">residual maximum likelihood</a> estimation of the variance components.

<div class="mw-heading mw-heading3"><h3 id="Partial_likelihood">Partial likelihood</h3></div>

A partial likelihood is an adaption of the full likelihood such that only a part of the parameters (the parameters of interest) occur in it.<a href="#cite_note-33">[33]</a> It is a key component of the <a href="Proportional_hazards_model" title="Proportional hazards model">proportional hazards model</a>: using a restriction on the hazard function, the likelihood does not contain the shape of the hazard over time.

<div class="mw-heading mw-heading2"><h2 id="Products_of_likelihoods">Products of likelihoods</h2></div>

The likelihood, given two or more <a href="Independence_(probability_theory)" title="Independence (probability theory)">independent</a> <a href="Event_(probability_theory)" title="Event (probability theory)">events</a>, is the product of the likelihoods of each of the individual events:

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (A\mid X_{1}\land X_{2})=\Lambda (A\mid X_{1})\cdot \Lambda (A\mid X_{2}).}</annotation>

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This follows from the definition of independence in probability: the probabilities of two independent events happening, given a model, is the product of the probabilities.

This is particularly important when the events are from <a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed random variables</a>, such as independent observations or <a href="Sampling_with_replacement" class="mw-redirect" title="Sampling with replacement">sampling with replacement</a>. In such a situation, the likelihood function factors into a product of individual likelihood functions.

The empty product has value 1, which corresponds to the likelihood, given no event, being 1: before any data, the likelihood is always 1. This is similar to a <a href="Uniform_prior" class="mw-redirect" title="Uniform prior">uniform prior</a> in Bayesian statistics, but in likelihoodist statistics this is not an <a href="Improper_prior" class="mw-redirect" title="Improper prior">improper prior</a> because likelihoods are not integrated.

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

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

Log-likelihood function is the logarithm of the likelihood function, often denoted by a lowercase l or ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }">

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

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</math><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" loading="lazy">⁠, to contrast with the uppercase L or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {L}}}">

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<annotation encoding="application/x-tex">{\textstyle {\mathcal {L}}}</annotation>

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</math><img src="./9e97aae95c92641d7ecba829fa92d6fe23888465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\textstyle {\mathcal {L}}}" loading="lazy"> for the likelihood. Because logarithms are <a href="Strictly_increasing" class="mw-redirect" title="Strictly increasing">strictly increasing</a> functions, maximizing the likelihood is equivalent to maximizing the log-likelihood. But for practical purposes it is more convenient to work with the log-likelihood function in <a href="Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimation</a>, in particular since most common <a href="Probability_distribution" title="Probability distribution">probability distributions</a>—notably the <a href="Exponential_family" title="Exponential family">exponential family</a>—are only <a href="Logarithmically_concave_function" title="Logarithmically concave function">logarithmically concave</a>,<a href="#cite_note-34">[34]</a><a href="#cite_note-35">[35]</a> and <a href="Concave_function" title="Concave function">concavity</a> of the <a href="Objective_function" class="mw-redirect" title="Objective function">objective function</a> plays a key role in the <a href="Mathematical_optimization" title="Mathematical optimization">maximization</a>.

Given the independence of each event, the overall log-likelihood of intersection equals the sum of the log-likelihoods of the individual events. This is analogous to the fact that the overall <a href="Log-probability" class="mw-redirect" title="Log-probability">log-probability</a> is the sum of the log-probability of the individual events. In addition to the mathematical convenience from this, the adding process of log-likelihood has an intuitive interpretation, as often expressed as "support" from the data. When the parameters are estimated using the log-likelihood for the <a href="Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimation</a>, each data point is used by being added to the total log-likelihood. As the data can be viewed as an evidence that support the estimated parameters, this process can be interpreted as "support from independent evidence adds", and the log-likelihood is the "weight of evidence". Interpreting negative log-probability as <a href="Information_content" title="Information content">information content</a> or <a href="Surprisal" class="mw-redirect" title="Surprisal">surprisal</a>, the support (log-likelihood) of a model, given an event, is the negative of the surprisal of the event, given the model: a model is supported by an event to the extent that the event is unsurprising, given the model.

A logarithm of a likelihood ratio is equal to the difference of the log-likelihoods:

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<annotation encoding="application/x-tex">{\displaystyle \log {\frac {{\mathcal {L}}(A)}{{\mathcal {L}}(B)}}=\log {\mathcal {L}}(A)-\log {\mathcal {L}}(B)=\ell (A)-\ell (B).}</annotation>

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Just as the likelihood, given no event, being 1, the log-likelihood, given no event, is 0, which corresponds to the value of the empty sum: without any data, there is no support for any models.

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

The <a href="Graph_of_a_function" title="Graph of a function">graph</a> of the log-likelihood is called the support curve (in the <a href="Univariate" title="Univariate">univariate</a> case).<a href="#cite_note-Edwards72-36">[36]</a>

In the multivariate case, the concept generalizes into a support surface over the <a href="Parameter_space" title="Parameter space">parameter space</a>.

It has a relation to, but is distinct from, the <a href="Support_(mathematics)#Support_(statistics)" title="Support (mathematics)">support of a distribution</a>.

The term was coined by <a href="A._W._F._Edwards" title="A. W. F. Edwards">A. W. F. Edwards</a><a href="#cite_note-Edwards72-36">[36]</a> in the context of <a href="Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">statistical hypothesis testing</a>, i.e. whether or not the data "support" one hypothesis (or parameter value) being tested more than any other.

The log-likelihood function being plotted is used in the computation of the <a href="Score_(statistics)" class="mw-redirect" title="Score (statistics)">score</a> (the gradient of the log-likelihood) and <a href="Fisher_information" title="Fisher information">Fisher information</a> (the curvature of the log-likelihood). Thus, the graph has a direct interpretation in the context of <a href="Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimation</a> and <a href="Likelihood-ratio_test" title="Likelihood-ratio test">likelihood-ratio tests</a>.

<div class="mw-heading mw-heading3"><h3 id="Likelihood_equations">Likelihood equations</h3></div>

If the log-likelihood function is <a href="Smoothness" title="Smoothness">smooth</a>, its <a href="Gradient" title="Gradient">gradient</a> with respect to the parameter, known as the <a href="Score_(statistics)" class="mw-redirect" title="Score (statistics)">score</a> and written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s_{n}(\theta )\equiv \nabla _{\theta }\ell _{n}(\theta )}">

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<annotation encoding="application/x-tex">{\textstyle s_{n}(\theta )\equiv \nabla _{\theta }\ell _{n}(\theta )}</annotation>

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The equations defined by the stationary point of the score function serve as <a href="Estimating_equations" title="Estimating equations">estimating equations</a> for the maximum likelihood estimator.

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In that sense, the maximum likelihood estimator is implicitly defined by the value at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {0} }">

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

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</math><img src="./688c89452cec92df9d11e5f4fd51cbb325aa9267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\textstyle \mathbf {0} }" loading="lazy"> of the <a href="Inverse_function" title="Inverse function">inverse function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s_{n}^{-1}:\mathbb {E} ^{d}\to \Theta }">

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle s_{n}^{-1}:\mathbb {E} ^{d}\to \Theta }</annotation>

</semantics>

</math><img src="./fec10262861dafd396b6d457afbc41e09b64401b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.425ex; height:3.009ex;" alt="{\textstyle s_{n}^{-1}:\mathbb {E} ^{d}\to \Theta }" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbb {E} ^{d}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbb {E} ^{d}}</annotation>

</semantics>

</math><img src="./49c4cf491c4fda80f67014510c6d7b0aa27fd028.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.676ex;" alt="{\textstyle \mathbb {E} ^{d}}" loading="lazy"> is the <var>d</var>-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4ce78034dc51a51642f247b9746264abd71704e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\textstyle \Theta }" loading="lazy"> is the parameter space. Using the <a href="Inverse_function_theorem" title="Inverse function theorem">inverse function theorem</a>, it can be shown that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s_{n}^{-1}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle s_{n}^{-1}}</annotation>

</semantics>

</math><img src="./91d376d9655612375e1a070abf0e244a962d771f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.423ex; height:3.009ex;" alt="{\textstyle s_{n}^{-1}}" loading="lazy"> is <a href="Well-defined" class="mw-redirect" title="Well-defined">well-defined</a> in an <a href="Open_neighborhood" class="mw-redirect" title="Open neighborhood">open neighborhood</a> about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {0} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./688c89452cec92df9d11e5f4fd51cbb325aa9267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\textstyle \mathbf {0} }" loading="lazy"> with probability going to one, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\hat {\theta }}_{n}=s_{n}^{-1}(\mathbf {0} )}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\hat {\theta }}_{n}=s_{n}^{-1}(\mathbf {0} )}</annotation>

</semantics>

</math><img src="./9877a1562c5757e6018853133c6f5733d476a56c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.242ex; height:3.343ex;" alt="{\textstyle {\hat {\theta }}_{n}=s_{n}^{-1}(\mathbf {0} )}" loading="lazy"> is a consistent estimate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.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="{\textstyle \theta }" loading="lazy">. As a consequence there exists a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{{\hat {\theta }}_{n}\right\}}">

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left\{{\hat {\theta }}_{n}\right\}}</annotation>

</semantics>

</math><img src="./d5233341cfc7df4c48855ecf30901a93e3792b43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.675ex; height:4.843ex;" alt="{\textstyle \left\{{\hat {\theta }}_{n}\right\}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s_{n}({\hat {\theta }}_{n})=\mathbf {0} }">

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle s_{n}({\hat {\theta }}_{n})=\mathbf {0} }</annotation>

</semantics>

</math><img src="./49bd0867e5a9b83c5b41580d1d93e408981107b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.128ex; height:3.343ex;" alt="{\textstyle s_{n}({\hat {\theta }}_{n})=\mathbf {0} }" loading="lazy"> asymptotically <a href="Almost_surely" title="Almost surely">almost surely</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\hat {\theta }}_{n}\xrightarrow {\text{p}} \theta _{0}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mover>

</mpadded>

</mover>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\hat {\theta }}_{n}\xrightarrow {\text{p}} \theta _{0}}</annotation>

</semantics>

</math><img src="./75fe4e0f661a0bac275d5f4db8e3353d84854ea3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.442ex; width:8.333ex; height:3.843ex;" alt="{\textstyle {\hat {\theta }}_{n}\xrightarrow {\text{p}} \theta _{0}}" loading="lazy">.<a href="#cite_note-37">[37]</a> A similar result can be established using <a href="Rolle's_theorem" title="Rolle's theorem">Rolle's theorem</a>.<a href="#cite_note-38">[38]</a><a href="#cite_note-39">[39]</a>

The second derivative evaluated at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\hat {\theta }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./12604986cd8d01bc09f476d5baafbfbcae2367af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\textstyle {\hat {\theta }}}" loading="lazy">, known as <a href="Fisher_information" title="Fisher information">Fisher information</a>, determines the curvature of the likelihood surface,<a href="#cite_note-40">[40]</a> and thus indicates the <a href="Precision_(statistics)" title="Precision (statistics)">precision</a> of the estimate.<a href="#cite_note-41">[41]</a>

<div class="mw-heading mw-heading3"><h3 id="Exponential_families">Exponential families</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Exponential_family" title="Exponential family">Exponential family</a></div>

The log-likelihood is also particularly useful for <a href="Exponential_families" class="mw-redirect" title="Exponential families">exponential families</a> of distributions, which include many of the common <a href="Parametric_model" title="Parametric model">parametric probability distributions</a>. The probability distribution function (and thus likelihood function) for exponential families contain products of factors involving <a href="Exponentiation" title="Exponentiation">exponentiation</a>. The logarithm of such a function is a sum of products, again easier to differentiate than the original function.

An exponential family is one whose probability density function is of the form (for some functions, writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \langle -,-\rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \langle -,-\rangle }</annotation>

</semantics>

</math><img src="./36f9f678068a39f20b78c429c2d832e12a114957.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\textstyle \langle -,-\rangle }" loading="lazy"> for the <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a>):

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x\mid {\boldsymbol {\theta }})=h(x)\exp {\Big (}\langle {\boldsymbol {\eta }}({\boldsymbol {\theta }}),\mathbf {T} (x)\rangle -A({\boldsymbol {\theta }}){\Big )}.}</annotation>

</semantics>

</math>

Each of these terms has an interpretation,<a href="#cite_note-42">[a]</a> but simply switching from probability to likelihood and taking logarithms yields the sum:

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ({\boldsymbol {\theta }}\mid x)=\langle {\boldsymbol {\eta }}({\boldsymbol {\theta }}),\mathbf {T} (x)\rangle -A({\boldsymbol {\theta }})+\log h(x).}</annotation>

</semantics>

</math>

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\boldsymbol {\eta }}({\boldsymbol {\theta }})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\boldsymbol {\eta }}({\boldsymbol {\theta }})}</annotation>

</semantics>

</math><img src="./5ebcd9e0cb49caf50d75552936770283d609e14b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.51ex; height:2.843ex;" alt="{\textstyle {\boldsymbol {\eta }}({\boldsymbol {\theta }})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle h(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle h(x)}</annotation>

</semantics>

</math><img src="./ac3e9aa522d8185d284098257119547e4ad5acf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.478ex; height:2.843ex;" alt="{\textstyle h(x)}" loading="lazy"> each correspond to a <a href="Change_of_coordinates" class="mw-redirect" title="Change of coordinates">change of coordinates</a>, so in these coordinates, the log-likelihood of an exponential family is given by the simple formula:

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ({\boldsymbol {\eta }}\mid x)=\langle {\boldsymbol {\eta }},\mathbf {T} (x)\rangle -A({\boldsymbol {\eta }}).}</annotation>

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</math>

In words, the log-likelihood of an exponential family is inner product of the natural parameter ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\eta }}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4918a87b7c00527526e2c705509100f5c4968f97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.395ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\eta }}}" loading="lazy">⁠ and the <a href="Sufficient_statistic" title="Sufficient statistic">sufficient statistic</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} (x)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e48e9ae6bc1c5d5c6fc7b2c202333edeb4ce8f25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.998ex; height:2.843ex;" alt="{\displaystyle \mathbf {T} (x)}" loading="lazy">⁠, minus the normalization factor (<a href="Log-partition_function" class="mw-redirect" title="Log-partition function">log-partition function</a>) ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A({\boldsymbol {\eta }})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A({\boldsymbol {\eta }})}</annotation>

</semantics>

</math><img src="./2521daf07a26fd34b8804b1484e1f103d4b99290.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle A({\boldsymbol {\eta }})}" loading="lazy">⁠. Thus for example the maximum likelihood estimate can be computed by taking derivatives of the sufficient statistic T and the log-partition function A.

<div class="mw-heading mw-heading4"><h4 id="Example:_the_gamma_distribution">Example: the gamma distribution</h4></div>

The <a href="Gamma_distribution" title="Gamma distribution">gamma distribution</a> is an exponential family with two parameters, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0d86dbd6183264b2f8569da1751380b173c7b185.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="{\textstyle \alpha }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a77bfb138e56b5b44f6c8c4ce32a05449d1573d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\textstyle \beta }" loading="lazy">. The likelihood function is

</mrow>

<mfrac>

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(\alpha ,\beta \mid x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}.}</annotation>

</semantics>

</math>

Finding the maximum likelihood estimate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a77bfb138e56b5b44f6c8c4ce32a05449d1573d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\textstyle \beta }" loading="lazy"> for a single observed value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.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="{\textstyle x}" loading="lazy"> looks rather daunting. Its logarithm is much simpler to work with:

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \log {\mathcal {L}}(\alpha ,\beta \mid x)=\alpha \log \beta -\log \Gamma (\alpha )+(\alpha -1)\log x-\beta x.\,}</annotation>

</semantics>

</math>

To maximize the log-likelihood, we first take the <a href="Partial_derivative" title="Partial derivative">partial derivative</a> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a77bfb138e56b5b44f6c8c4ce32a05449d1573d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\textstyle \beta }" loading="lazy">:

<mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \mid x)}{\partial \beta }}={\frac {\alpha }{\beta }}-x.}</annotation>

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</math>

If there are a number of independent observations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{1},\ldots ,x_{n}}">

<msub>

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<msub>

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

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</math><img src="./3e9531d09966e9ceeb357705fc047d0c907d3841.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="{\textstyle x_{1},\ldots ,x_{n}}" loading="lazy">, then the joint log-likelihood will be the sum of individual log-likelihoods, and the derivative of this sum will be a sum of derivatives of each individual log-likelihood:

<mtr>

<mtd>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \mid x_{1},\ldots ,x_{n})}{\partial \beta }}\\&={\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \mid x_{1})}{\partial \beta }}+\cdots +{\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \mid x_{n})}{\partial \beta }}\\&={\frac {n\alpha }{\beta }}-\sum _{i=1}^{n}x_{i}.\end{aligned}}}</annotation>

</semantics>

</math>

To complete the maximization procedure for the joint log-likelihood, the equation is set to zero and solved for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a77bfb138e56b5b44f6c8c4ce32a05449d1573d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\textstyle \beta }" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}={\frac {\alpha }{\bar {x}}}.}</annotation>

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Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\widehat {\beta }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\widehat {\beta }}}</annotation>

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</math><img src="./3089c659b3a7782bc4ebca0a58858433803f66a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\textstyle {\widehat {\beta }}}" loading="lazy"> denotes the maximum-likelihood estimate, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}">

<mover>

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<annotation encoding="application/x-tex">{\textstyle \textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}</annotation>

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</math><img src="./3392954e18808cba0f9572e641ceee865bc225e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.508ex; height:3.343ex;" alt="{\textstyle \textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}" loading="lazy"> is the <a href="Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a> of the observations.

<div class="mw-heading mw-heading2"><h2 id="Background_and_interpretation">Background and interpretation</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Historical_remarks">Historical remarks</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="History_of_statistics" title="History of statistics">History of statistics</a> and <a href="History_of_probability" title="History of probability">History of probability</a></div>

The term "likelihood" has been in use in English since at least late <a href="Middle_English" title="Middle English">Middle English</a>.<a href="#cite_note-43">[42]</a> Its formal use to refer to a specific <a href="Function_(mathematics)" title="Function (mathematics)">function</a> in mathematical statistics was proposed by <a href="Ronald_Fisher" title="Ronald Fisher">Ronald Fisher</a>,<a href="#cite_note-44">[43]</a> in two research papers published in 1921<a href="#cite_note-45">[44]</a> and 1922.<a href="#cite_note-Fisher1922-46">[45]</a> The 1921 paper introduced what is today called a "likelihood interval"; the 1922 paper introduced the term "<a href="Method_of_maximum_likelihood" class="mw-redirect" title="Method of maximum likelihood">method of maximum likelihood</a>". Quoting Fisher:

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</style><blockquote class="templatequote">[I]n 1922, I proposed the term 'likelihood,' in view of the fact that, with respect to [the parameter], it is not a probability, and does not obey the laws of probability, while at the same time it bears to the problem of rational choice among the possible values of [the parameter] a relation similar to that which probability bears to the problem of predicting events in games of chance. . . . Whereas, however, in relation to psychological judgment, likelihood has some resemblance to probability, the two concepts are wholly distinct. . . ."<a href="#cite_note-47">[46]</a></blockquote>

The concept of likelihood should not be confused with probability as mentioned by Sir Ronald Fisher

<blockquote class="templatequote">I stress this because in spite of the emphasis that I have always laid upon the difference between probability and likelihood there is still a tendency to treat likelihood as though it were a sort of probability. The first result is thus that there are two different measures of rational belief appropriate to different cases. Knowing the population we can express our incomplete knowledge of, or expectation of, the sample in terms of probability; knowing the sample we can express our incomplete knowledge of the population in terms of likelihood.<a href="#cite_note-48">[47]</a></blockquote>

Fisher's invention of statistical likelihood was in reaction against an earlier form of reasoning called <a href="Inverse_probability" title="Inverse probability">inverse probability</a>.<a href="#cite_note-49">[48]</a> His use of the term "likelihood" fixed the meaning of the term within mathematical statistics.

<a href="A._W._F._Edwards" title="A. W. F. Edwards">A. W. F. Edwards</a> (1972) established the axiomatic basis for use of the log-likelihood ratio as a measure of relative support for one hypothesis against another. The support function is then the natural logarithm of the likelihood function. Both terms are used in <a href="Phylogenetics" title="Phylogenetics">phylogenetics</a>, but were not adopted in a general treatment of the topic of statistical evidence.<a href="#cite_note-50">[49]</a>

<div class="mw-heading mw-heading3"><h3 id="Interpretations_under_different_foundations">Interpretations under different foundations</h3></div>

Among statisticians, there is no consensus about what the <a href="Foundations_of_statistics" title="Foundations of statistics">foundation of statistics</a> should be. There are four main paradigms that have been proposed for the foundation: <a href="Frequentism" class="mw-redirect" title="Frequentism">frequentism</a>, <a href="Bayesianism" class="mw-redirect" title="Bayesianism">Bayesianism</a>, <a href="Likelihoodism" class="mw-redirect" title="Likelihoodism">likelihoodism</a>, and <a href="Akaike_information_criterion" title="Akaike information criterion">AIC-based</a>.<a href="#cite_note-BF11-51">[50]</a> For each of the proposed foundations, the interpretation of likelihood is different. The four interpretations are described in the subsections below.

<div class="mw-heading mw-heading4"><h4 id="Frequentist_interpretation">Frequentist interpretation</h4></div>

<div class="mw-heading mw-heading4"><h4 id="Bayesian_interpretation">Bayesian interpretation</h4></div>

In <a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</a>, although one can speak about the likelihood of any proposition or <a href="Random_variable" title="Random variable">random variable</a> given another random variable: for example the likelihood of a parameter value or of a <a href="Statistical_model" title="Statistical model">statistical model</a> (see <a href="Marginal_likelihood" title="Marginal likelihood">marginal likelihood</a>), given specified data or other evidence,<a href="#cite_note-good1950-52">[51]</a><a href="#cite_note-jeffreys1983-53">[52]</a><a href="#cite_note-jaynes2003-54">[53]</a><a href="#cite_note-lindley1980-55">[54]</a> the likelihood function remains the same entity, with the additional interpretations of (i) a <a href="Conditional_probability_distribution" title="Conditional probability distribution">conditional density</a> of the data given the parameter (since the parameter is then a random variable) and (ii) a measure or amount of information brought by the data about the parameter value or even the model.<a href="#cite_note-good1950-52">[51]</a><a href="#cite_note-jeffreys1983-53">[52]</a><a href="#cite_note-jaynes2003-54">[53]</a><a href="#cite_note-lindley1980-55">[54]</a><a href="#cite_note-gelmanetal2014-56">[55]</a> Due to the introduction of a probability structure on the parameter space or on the collection of models, it is possible that a parameter value or a statistical model have a large likelihood value for given data, and yet have a low probability, or vice versa.<a href="#cite_note-jaynes2003-54">[53]</a><a href="#cite_note-gelmanetal2014-56">[55]</a> This is often the case in medical contexts.<a href="#cite_note-57">[56]</a> Following <a href="Bayes'_Rule" class="mw-redirect" title="Bayes' Rule">Bayes' Rule</a>, the likelihood when seen as a conditional density can be multiplied by the <a href="Prior_probability" title="Prior probability">prior probability</a> density of the parameter and then normalized, to give a <a href="Posterior_probability" title="Posterior probability">posterior probability</a> density.<a href="#cite_note-good1950-52">[51]</a><a href="#cite_note-jeffreys1983-53">[52]</a><a href="#cite_note-jaynes2003-54">[53]</a><a href="#cite_note-lindley1980-55">[54]</a><a href="#cite_note-gelmanetal2014-56">[55]</a> More generally, the likelihood of an unknown quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8d80c41192705e1a6c6de1d65e16d7f70fbac391.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="{\textstyle X}" loading="lazy"> given another unknown quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6222becc4f0c5effa012e5335b170575fdbbaad3.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="{\textstyle Y}" loading="lazy"> is proportional to the probability of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Y}">

</mstyle>

</mrow>

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

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</math><img src="./6222becc4f0c5effa012e5335b170575fdbbaad3.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="{\textstyle Y}" loading="lazy"> given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}">

</mstyle>

</mrow>

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

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</math><img src="./8d80c41192705e1a6c6de1d65e16d7f70fbac391.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="{\textstyle X}" loading="lazy">.<a href="#cite_note-good1950-52">[51]</a><a href="#cite_note-jeffreys1983-53">[52]</a><a href="#cite_note-jaynes2003-54">[53]</a><a href="#cite_note-lindley1980-55">[54]</a><a href="#cite_note-gelmanetal2014-56">[55]</a>

<div class="mw-heading mw-heading4"><h4 id="Likelihoodist_interpretation">Likelihoodist interpretation</h4></div>

In frequentist statistics, the likelihood function is itself a <a href="Statistic" title="Statistic">statistic</a> that summarizes a single sample from a population, whose calculated value depends on a choice of several parameters θ1 ... θp, where p is the count of parameters in some already-selected <a href="Statistical_model" title="Statistical model">statistical model</a>. The value of the likelihood serves as a figure of merit for the choice used for the parameters, and the parameter set with maximum likelihood is the best choice, given the data available.

The specific calculation of the likelihood is the probability that the observed sample would be assigned, assuming that the model chosen and the values of the several parameters θ give an accurate approximation of the <a href="Frequency_distribution" class="mw-redirect" title="Frequency distribution">frequency distribution</a> of the population that the observed sample was drawn from. Heuristically, it makes sense that a good choice of parameters is those which render the sample actually observed the maximum possible post-hoc probability of having happened. <a href="Wilks'_theorem" title="Wilks' theorem">Wilks' theorem</a> quantifies the heuristic rule by showing that the difference in the logarithm of the likelihood generated by the estimate's parameter values and the logarithm of the likelihood generated by population's "true" (but unknown) parameter values is asymptotically <a href="Chi-squared_distribution" title="Chi-squared distribution">χ2 distributed</a>.

Each independent sample's maximum likelihood estimate is a separate estimate of the "true" parameter set describing the population sampled. Successive estimates from many independent samples will cluster together with the population's "true" set of parameter values hidden somewhere in their midst. The difference in the logarithms of the maximum likelihood and adjacent parameter sets' likelihoods may be used to draw a <a href="Confidence_region" title="Confidence region">confidence region</a> on a plot whose co-ordinates are the parameters θ1 ... θp. The region surrounds the maximum-likelihood estimate, and all points (parameter sets) within that region differ at most in log-likelihood by some fixed value. The <a href="Chi-squared_distribution" title="Chi-squared distribution">χ2 distribution</a> given by <a href="Wilks'_theorem" title="Wilks' theorem">Wilks' theorem</a> converts the region's log-likelihood differences into the "confidence" that the population's "true" parameter set lies inside. The art of choosing the fixed log-likelihood difference is to make the confidence acceptably high while keeping the region acceptably small (narrow range of estimates).

As more data are observed, instead of being used to make independent estimates, they can be combined with the previous samples to make a single combined sample, and that large sample may be used for a new maximum likelihood estimate. As the size of the combined sample increases, the size of the likelihood region with the same confidence shrinks. Eventually, either the size of the confidence region is very nearly a single point, or the entire population has been sampled; in both cases, the estimated parameter set is essentially the same as the population parameter set.

<div class="mw-heading mw-heading4"><h4 id="AIC-based_interpretation">AIC-based interpretation</h4></div>

Under the <a href="Akaike_information_criterion" title="Akaike information criterion">AIC</a> paradigm, likelihood is interpreted within the context of <a href="Information_theory" title="Information theory">information theory</a>.<a href="#cite_note-58">[57]</a><a href="#cite_note-59">[58]</a><a href="#cite_note-60">[59]</a>

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

<li><a href="Conditional_entropy" title="Conditional entropy">Conditional entropy</a></li>

<li><a href="Conditional_probability" title="Conditional probability">Conditional probability</a></li>

<li><a href="Empirical_likelihood" title="Empirical likelihood">Empirical likelihood</a></li>

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

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

<li><a href="Likelihoodist_statistics" title="Likelihoodist statistics">Likelihoodist statistics</a></li>

<li><a href="Maximum_likelihood_estimation" title="Maximum likelihood estimation">Maximum likelihood estimation</a></li>

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

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

<li><a href="Score_(statistics)" class="mw-redirect" title="Score (statistics)">Score (statistics)</a></li></ul></div>

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

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<li id="cite_note-42"><a href="#cite_ref-42">^</a> See <a href="Exponential_family#Interpretation" title="Exponential family">Exponential family § Interpretation</a>

</li>

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

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

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<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFChen1985" class="citation journal cs1">Chen, Chan-Fu (1985). "On Asymptotic Normality of Limiting Density Functions with Bayesian Implications". Journal of the Royal Statistical Society. Series B (Methodological). 47 (3): 540–546. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.2517-6161.1985.tb01384.x">10.1111/j.2517-6161.1985.tb01384.x</a>.</cite>

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<li id="cite_note-Kalbfleisch-16">^ <a href="#cite_ref-Kalbfleisch_16-0">a</a> <a href="#cite_ref-Kalbfleisch_16-1">b</a> <a href="#cite_ref-Kalbfleisch_16-2">c</a> <a href="#cite_ref-Kalbfleisch_16-3">d</a> <cite id="CITEREFKalbfleisch1985" class="citation cs2"><a href="James_G._Kalbfleisch" title="James G. Kalbfleisch">Kalbfleisch, J. G.</a> (1985), Probability and Statistical Inference, Springer</cite> (§9.3).

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<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFAzzalini1996" class="citation cs2">Azzalini, A. (1996), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hyN6gXHvSo0C">Statistical Inference—Based on the likelihood</a>, <a href="Chapman_%26_Hall" title="Chapman & Hall">Chapman & Hall</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780412606502</bdi></cite> (§1.4.2).

</li>

<li id="cite_note-Sprott-18">^ <a href="#cite_ref-Sprott_18-0">a</a> <a href="#cite_ref-Sprott_18-1">b</a> <a href="#cite_ref-Sprott_18-2">c</a> Sprott, D. A. (2000), Statistical Inference in Science, Springer (chap. 2).

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> Davison, A. C. (2008), Statistical Models, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a> (§4.1.2).

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite id="CITEREFHeldSabanés_Bové2014" class="citation cs2">Held, L.; Sabanés Bové, D. S. (2014), Applied Statistical Inference—Likelihood and Bayes, Springer</cite> (§2.1).

</li>

<li id="cite_note-Rossi2018-21">^ <a href="#cite_ref-Rossi2018_21-0">a</a> <a href="#cite_ref-Rossi2018_21-1">b</a> <a href="#cite_ref-Rossi2018_21-2">c</a> <cite id="CITEREFRossi2018" class="citation cs2">Rossi, R. J. (2018), Mathematical Statistics, <a href="Wiley_(publisher)" title="Wiley (publisher)">Wiley</a>, p. 267</cite>.

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<li id="cite_note-Hudson-22">^ <a href="#cite_ref-Hudson_22-0">a</a> <a href="#cite_ref-Hudson_22-1">b</a> <cite id="CITEREFHudson1971" class="citation cs2">Hudson, D. J. (1971), "Interval estimation from the likelihood function", <a href="Journal_of_the_Royal_Statistical_Society%2C_Series_B" class="mw-redirect" title="Journal of the Royal Statistical Society, Series B">Journal of the Royal Statistical Society, Series B</a>, 33 (2): 256–262, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.2517-6161.1971.tb00877.x">10.1111/j.2517-6161.1971.tb00877.x</a></cite>.

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFPawitan2001" class="citation book cs1">Pawitan, Yudi (2001). <a rel="nofollow" class="external text" href="https://archive.org/details/inalllikelihood0000unse">In All Likelihood: Statistical Modelling and Inference Using Likelihood</a>. <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFWen_Hsiang_Wei" class="citation web cs1">Wen Hsiang Wei. <a rel="nofollow" class="external text" href="http://web.thu.edu.tw/wenwei/www/glmpdfmargin.htm">"Generalized Linear Model - course notes"</a>. Taichung, Taiwan: <a href="Tunghai_University" title="Tunghai University">Tunghai University</a>. pp. Chapter 5. Retrieved 2017-10-01.</cite>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFAmemiya1985" class="citation book cs1"><a href="Takeshi_Amemiya" title="Takeshi Amemiya">Amemiya, Takeshi</a> (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0bzGQE14CwEC&pg=PA125">"Concentrated Likelihood Function"</a>. <a rel="nofollow" class="external text" href="https://archive.org/details/advancedeconomet00amem/page/125">Advanced Econometrics</a>. Cambridge: Harvard University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/advancedeconomet00amem/page/125">125–127</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-674-00560-0</bdi>.</cite>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFDavidsonMacKinnon1993" class="citation book cs1">Davidson, Russell; <a href="James_G._MacKinnon" title="James G. MacKinnon">MacKinnon, James G.</a> (1993). "Concentrating the Loglikelihood Function". Estimation and Inference in Econometrics. New York: Oxford University Press. pp. 267–269. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-506011-9</bdi>.</cite>

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<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFGourierouxMonfort1995" class="citation book cs1">Gourieroux, Christian; Monfort, Alain (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gqI-pAP2JZ8C&pg=PA170">"Concentrated Likelihood Function"</a>. Statistics and Econometric Models. New York: Cambridge University Press. pp. 170–175. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-40551-5</bdi>.</cite>

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<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite id="CITEREFPickles1985" class="citation cs1">Pickles, Andrew (1985). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoli0000pick/page/21">An Introduction to Likelihood Analysis</a>. Norwich: W. H. Hutchins & Sons. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoli0000pick/page/21">21–24</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-86094-190-6</bdi>.</cite>

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<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite id="CITEREFBolker2008" class="citation book cs1">Bolker, Benjamin M. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=flyBd1rpqeoC&pg=PA188">Ecological Models and Data in R</a>. Princeton University Press. pp. 187–189. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-12522-0</bdi>.</cite>

</li>

<li id="cite_note-30"><a href="#cite_ref-30">^</a> <cite id="CITEREFAitkin1982" class="citation cs1"><a href="Murray_Aitkin" title="Murray Aitkin">Aitkin, Murray</a> (1982). "Direct Likelihood Inference". GLIM 82: Proceedings of the International Conference on Generalised Linear Models. Springer. pp. 76–86. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-90777-7</bdi>.</cite>

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<li id="cite_note-31"><a href="#cite_ref-31">^</a> <cite id="CITEREFVenzonMoolgavkar1988" class="citation cs1">Venzon, D. J.; Moolgavkar, S. H. (1988). "A Method for Computing Profile-Likelihood-Based Confidence Intervals". <a href="Journal_of_the_Royal_Statistical_Society" title="Journal of the Royal Statistical Society">Journal of the Royal Statistical Society</a>. Series C (Applied Statistics). 37 (1): 87–94. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2347496">10.2307/2347496</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2347496">2347496</a>.</cite>

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<li id="cite_note-32"><a href="#cite_ref-32">^</a> <cite id="CITEREFKalbfleischSprott1973" class="citation journal cs1">Kalbfleisch, J. D.; Sprott, D. A. (1973). "Marginal and Conditional Likelihoods". Sankhyā: The Indian Journal of Statistics. Series A. 35 (3): 311–328. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/25049882">25049882</a>.</cite>

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<cite id="CITEREFCox1975" class="citation cs1"><a href="David_Cox_(statistician)" title="David Cox (statistician)">Cox, D. R.</a> (1975). "Partial likelihood". <a href="Biometrika" title="Biometrika">Biometrika</a>. 62 (2): 269–276. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F62.2.269">10.1093/biomet/62.2.269</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=0400509">0400509</a>.</cite>

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<li id="cite_note-34"><a href="#cite_ref-34">^</a> <cite id="CITEREFKassVos1997" class="citation cs1">Kass, Robert E.; Vos, Paul W. (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e43EAIfUPCwC&pg=PA14">Geometrical Foundations of Asymptotic Inference</a>. New York: John Wiley & Sons. p. 14. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-82668-5</bdi>.</cite>

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<li id="cite_note-35"><a href="#cite_ref-35">^</a> <cite id="CITEREFPapadopoulos2013" class="citation web cs1">Papadopoulos, Alecos (September 25, 2013). <a rel="nofollow" class="external text" href="https://stats.stackexchange.com/q/70975">"Why we always put log() before the joint pdf when we use MLE (Maximum likelihood Estimation)?"</a>. <a href="Stack_Exchange" title="Stack Exchange">Stack Exchange</a>.</cite>

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<li id="cite_note-Edwards72-36">^ <a href="#cite_ref-Edwards72_36-0">a</a> <a href="#cite_ref-Edwards72_36-1">b</a> <cite id="CITEREFEdwards1992" class="citation book cs1"><a href="A._W._F._Edwards" title="A. W. F. Edwards">Edwards, A. W. F.</a> (1992) [1972]. Likelihood. <a href="Johns_Hopkins_University_Press" title="Johns Hopkins University Press">Johns Hopkins University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8018-4443-6</bdi>.</cite>

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<li id="cite_note-37"><a href="#cite_ref-37">^</a> <cite id="CITEREFFoutz1977" class="citation journal cs1">Foutz, Robert V. (1977). "On the Unique Consistent Solution to the Likelihood Equations". <a href="Journal_of_the_American_Statistical_Association" title="Journal of the American Statistical Association">Journal of the American Statistical Association</a>. 72 (357): 147–148. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1977.10479926">10.1080/01621459.1977.10479926</a>.</cite>

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<li id="cite_note-38"><a href="#cite_ref-38">^</a> <cite id="CITEREFTaroneGruenhage1975" class="citation journal cs1">Tarone, Robert E.; Gruenhage, Gary (1975). "A Note on the Uniqueness of Roots of the Likelihood Equations for Vector-Valued Parameters". Journal of the American Statistical Association. 70 (352): 903–904. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1975.10480321">10.1080/01621459.1975.10480321</a>.</cite>

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<li id="cite_note-39"><a href="#cite_ref-39">^</a> <cite id="CITEREFRaiVan_Ryzin1982" class="citation journal cs1">Rai, Kamta; Van Ryzin, John (1982). "A Note on a Multivariate Version of Rolle's Theorem and Uniqueness of Maximum Likelihood Roots". Communications in Statistics. Theory and Methods. 11 (13): 1505–1510. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03610928208828325">10.1080/03610928208828325</a>.</cite>

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<li id="cite_note-40"><a href="#cite_ref-40">^</a> <cite id="CITEREFRao1960" class="citation cs1">Rao, B. Raja (1960). "A formula for the curvature of the likelihood surface of a sample drawn from a distribution admitting sufficient statistics". <a href="Biometrika" title="Biometrika">Biometrika</a>. 47 (1–2): 203–207. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F47.1-2.203">10.1093/biomet/47.1-2.203</a>.</cite>

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<li id="cite_note-41"><a href="#cite_ref-41">^</a> <cite id="CITEREFWardAhlquist2018" class="citation cs1">Ward, Michael D.; Ahlquist, John S. (2018). Maximum Likelihood for Social Science : Strategies for Analysis. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. 25–27.</cite>

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<li id="cite_note-43"><a href="#cite_ref-43">^</a> "likelihood", <a href="Shorter_Oxford_English_Dictionary" title="Shorter Oxford English Dictionary">Shorter Oxford English Dictionary</a> (2007).

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<li id="cite_note-44"><a href="#cite_ref-44">^</a> <cite id="CITEREFHald1999" class="citation cs1"><a href="Anders_Hald" title="Anders Hald">Hald, A.</a> (1999). <a rel="nofollow" class="external text" href="http://projecteuclid.org/download/pdf_1/euclid.ss/1009212248">"On the history of maximum likelihood in relation to inverse probability and least squares"</a>. <a href="Statistical_Science" title="Statistical Science">Statistical Science</a>. 14 (2): 214–222. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1009212248">10.1214/ss/1009212248</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2676741">2676741</a>.</cite>

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<li id="cite_note-45"><a href="#cite_ref-45">^</a> <cite id="CITEREFFisher1921" class="citation cs1"><a href="Ronald_Fisher" title="Ronald Fisher">Fisher, R.A.</a> (1921). "On the "probable error" of a coefficient of correlation deduced from a small sample". Metron. 1: 3–32.</cite>

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<li id="cite_note-Fisher1922-46"><a href="#cite_ref-Fisher1922_46-0">^</a> <cite id="CITEREFFisher1922" class="citation cs1"><a href="Ronald_Fisher" title="Ronald Fisher">Fisher, R.A.</a> (1922). <a rel="nofollow" class="external text" href="http://digital.library.adelaide.edu.au/dspace/handle/2440/15172">"On the mathematical foundations of theoretical statistics"</a>. Philosophical Transactions of the Royal Society A. 222 (594–604): 309–368. <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/1922RSPTA.222..309F">1922RSPTA.222..309F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1922.0009">10.1098/rsta.1922.0009</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2440%2F15172">2440/15172</a>. <a href="JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:48.1280.02">48.1280.02</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/91208">91208</a>.</cite>

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<li id="cite_note-47"><a href="#cite_ref-47">^</a> <cite id="CITEREFKlemens2008" class="citation cs1">Klemens, Ben (2008). Modeling with Data: Tools and Techniques for Scientific Computing. <a href="Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. p. 329.</cite>

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<li id="cite_note-48"><a href="#cite_ref-48">^</a> <cite id="CITEREFFisher1930" class="citation cs1"><a href="Ronald_Fisher" title="Ronald Fisher">Fisher, Ronald</a> (1930). "Inverse Probability". <a href="Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society" title="Mathematical Proceedings of the Cambridge Philosophical Society">Mathematical Proceedings of the Cambridge Philosophical Society</a>. 26 (4): 528–535. <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/1930PCPS...26..528F">1930PCPS...26..528F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004100016297">10.1017/S0305004100016297</a>.</cite>

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<li id="cite_note-49"><a href="#cite_ref-49">^</a> <cite id="CITEREFFienberg1997" class="citation cs1">Fienberg, Stephen E (1997). "Introduction to R.A. Fisher on inverse probability and likelihood". <a href="Statistical_Science" title="Statistical Science">Statistical Science</a>. 12 (3): 161. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1030037905">10.1214/ss/1030037905</a>.</cite>

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<li id="cite_note-50"><a href="#cite_ref-50">^</a> <cite id="CITEREFRoyall1997" class="citation cs1">Royall, R. (1997). Statistical Evidence. <a href="Chapman_%26_Hall" title="Chapman & Hall">Chapman & Hall</a>.</cite>

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<li id="cite_note-BF11-51"><a href="#cite_ref-BF11_51-0">^</a> <cite id="CITEREFBandyopadhyayForster2011" class="citation cs1">Bandyopadhyay, P. S.; Forster, M. R., eds. (2011). Philosophy of Statistics. <a href="North-Holland_Publishing" class="mw-redirect" title="North-Holland Publishing">North-Holland Publishing</a>.</cite>

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<li id="cite_note-good1950-52">^ <a href="#cite_ref-good1950_52-0">a</a> <a href="#cite_ref-good1950_52-1">b</a> <a href="#cite_ref-good1950_52-2">c</a> <a href="#cite_ref-good1950_52-3">d</a> I. J. Good: Probability and the Weighing of Evidence (Griffin 1950), §6.1

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<li id="cite_note-57"><a href="#cite_ref-57">^</a> <cite id="CITEREFSoxHigginsOwens2013" class="citation cs2">Sox, H. C.; Higgins, M. C.; Owens, D. K. (2013), Medical Decision Making (2nd ed.), Wiley, chapters 3–4, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9781118341544">10.1002/9781118341544</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781118341544</bdi></cite>

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<li id="cite_note-58"><a href="#cite_ref-58">^</a> <cite id="CITEREFAkaike1985" class="citation cs1"><a href="Hirotugu_Akaike" title="Hirotugu Akaike">Akaike, H.</a> (1985). "Prediction and entropy". In Atkinson, A. C.; <a href="Stephen_Fienberg" title="Stephen Fienberg">Fienberg, S. E.</a> (eds.). A Celebration of Statistics. Springer. pp. 1–24.</cite>

</li>

<li id="cite_note-59"><a href="#cite_ref-59">^</a> <cite id="CITEREFSakamotoIshiguroKitagawa1986" class="citation cs1">Sakamoto, Y.; Ishiguro, M.; Kitagawa, G. (1986). Akaike Information Criterion Statistics. <a href="D._Reidel" title="D. Reidel">D. Reidel</a>. Part I.</cite>

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<li id="cite_note-60"><a href="#cite_ref-60">^</a> <cite id="CITEREFBurnhamAnderson2002" class="citation cs1">Burnham, K. P.; Anderson, D. R. (2002). Model Selection and Multimodel Inference: A practical information-theoretic approach (2nd ed.). <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. chap. 7.</cite>

</li>

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

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

<ul><li><cite id="CITEREFAzzalini1996" class="citation cs1">Azzalini, Adelchi (1996). "Likelihood". Statistical Inference Based on the Likelihood. Chapman and Hall. pp. 17–50. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-412-60650-X</bdi>.</cite></li>

<li><cite id="CITEREFBoosStefanski2013" class="citation cs1">Boos, Dennis D.; Stefanski, L. A. (2013). "Likelihood Construction and Estimation". Essential Statistical Inference : Theory and Methods. New York: Springer. pp. 27–124. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4614-4818-1_2">10.1007/978-1-4614-4818-1_2</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4614-4817-4</bdi>.</cite></li>

<li><cite id="CITEREFEdwards1992" class="citation cs1"><a href="A._W._F._Edwards" title="A. W. F. Edwards">Edwards, A. W. F.</a> (1992) [1972]. Likelihood (Expanded ed.). <a href="Johns_Hopkins_University_Press" title="Johns Hopkins University Press">Johns Hopkins University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8018-4443-6</bdi>.</cite></li>

<li><cite id="CITEREFKing1989" class="citation cs1"><a href="Gary_King_(political_scientist)" title="Gary King (political scientist)">King, Gary</a> (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cligOwrd7XoC&pg=PA59">"The Likelihood Model of Inference"</a>. Unifying Political Methodology : the Likehood Theory of Statistical Inference. Cambridge University Press. pp. 59–94. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-36697-6</bdi>.</cite></li>

<li><cite id="CITEREFRichardVecer2021" class="citation journal cs1">Richard, Mark; Vecer, Jan (1 February 2021). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Frisks9020031">"Efficiency Testing of Prediction Markets: Martingale Approach, Likelihood Ratio and Bayes Factor Analysis"</a>. Risks. 9 (2): 31. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Frisks9020031">10.3390/risks9020031</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10419%2F258120">10419/258120</a>.</cite></li>

<li><cite id="CITEREFLindsey1996" class="citation cs1">Lindsey, J. K. (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/parametricstatis0000lind/page/69">"Likelihood"</a>. Parametric Statistical Inference. Oxford University Press. pp. 69–139. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-852359-9</bdi>.</cite></li>

<li><cite id="CITEREFRohde2014" class="citation cs1">Rohde, Charles A. (2014). Introductory Statistical Inference with the Likelihood Function. Berlin: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-10460-7</bdi>.</cite></li>

<li><cite id="CITEREFRoyall1997" class="citation cs1">Royall, Richard (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/statisticalevide0000roya">Statistical Evidence : A Likelihood Paradigm</a>. London: Chapman & Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-412-04411-0</bdi>.</cite></li>

<li><cite id="CITEREFWardAhlquist2018" class="citation cs1"><a href="Michael_D._Ward" title="Michael D. Ward">Ward, Michael D.</a>; Ahlquist, John S. (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iqRyDwAAQBAJ&pg=PA21">"The Likelihood Function: A Deeper Dive"</a>. Maximum Likelihood for Social Science : Strategies for Analysis. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. 21–28. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-316-63682-4</bdi>.</cite></li></ul>

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

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<div class="side-box-text plainlist">Look up <a href="https://en.wiktionary.org/wiki/likelihood" class="extiw external" title="wiktionary:likelihood">likelihood</a> in Wiktionary, the free dictionary.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="https://planetmath.org/likelihoodfunction">Likelihood function at Planetmath</a></li>

<li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.statlect.com/glossary/log-likelihood">"Log-likelihood"</a>. Statlect.</cite></li></ul>

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</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">Lp 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="Monotone_likelihood_ratio" title="Monotone likelihood ratio">(monotone)</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">Z-test (normal)</a></li>

<li><a href="Student's_t-test" title="Student's t-test">Student's t-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 (Shapiro–Wilk)</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 (Wilcoxon)</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 (Mann–Whitney)</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 (Kruskal–Wallis)</a></li>

<li><a href="Friedman_test" title="Friedman test">2-way (Friedman)</a></li>

<li><a href="Jonckheere's_trend_test" title="Jonckheere's trend test">Ordered alternative (Jonckheere–Terpstra)</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"><a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</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="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 (Bernoulli)</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 (Ljung–Box)</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 (Box–Jenkins)</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>

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