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<title>Recursive Bayesian estimation</title>

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

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about Bayes filter, a general probabilistic approach. For the spam filter with a similar name, see <a href="Naive_Bayes_spam_filtering" class="mw-redirect" title="Naive Bayes spam filtering">Naive Bayes spam filtering</a>.</div>

<p>In <a href="Probability_Theory" class="mw-redirect" title="Probability Theory">probability theory</a>, <a href="Statistics" title="Statistics">statistics</a>, and <a href="Machine_Learning" class="mw-redirect" title="Machine Learning">machine learning</a>, <b>recursive Bayesian estimation</b>, also known as a <b>Bayes filter</b>, is a general probabilistic approach for <a href="Density_estimation" title="Density estimation">estimating</a> an unknown <a href="Probability_density_function" title="Probability density function">probability density function</a> (<a href="Probability_density_function" title="Probability density function">PDF</a>) recursively over time using incoming measurements and a mathematical process model. The process relies heavily upon mathematical concepts and models that are theorized within a study of prior and posterior probabilities known as <a href="Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>.

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<div class="mw-heading mw-heading2"><h2 id="In_robotics">In robotics</h2></div>

<p>A Bayes filter is an algorithm used in <a href="Computer_science" title="Computer science">computer science</a> for calculating the probabilities of multiple beliefs to allow a <a href="Robot" title="Robot">robot</a> to infer its position and orientation. Essentially, Bayes filters allow robots to continuously update their most likely position within a coordinate system, based on the most recently acquired sensor data. This is a recursive algorithm. It consists of two parts: prediction and innovation. If the variables are <a href="Normal_Distribution" class="mw-redirect" title="Normal Distribution">normally distributed</a> and the transitions are linear, the Bayes filter becomes equal to the <a href="Kalman_filter" title="Kalman filter">Kalman filter</a>.

</p><p>In a simple example, a robot moving throughout a grid may have several different sensors that provide it with information about its surroundings. The robot may begin with certainty that it is at position (0,0). However, as it moves further and further from its original position, the robot has continuously less certainty about its position; using a Bayes filter, a probability can be assigned to the robot's belief about its current position, and that probability can be continuously updated from additional sensor information.

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

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

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</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span> are the <a href="Manifest_variable" class="mw-redirect" title="Manifest variable">manifestations</a> of a <a href="Hidden_Markov_model" title="Hidden Markov model">hidden Markov model</a> (HMM), which means the true state <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> is assumed to be an unobserved <a href="Markov_process" class="mw-redirect" title="Markov process">Markov process</a>. The following picture presents a <a href="Bayesian_network" title="Bayesian network">Bayesian network</a> of a HMM.

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<p>Because of the Markov assumption, the probability of the current true state given the immediately previous one is conditionally independent of the other earlier states.

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<annotation encoding="application/x-tex">{\displaystyle p({\textbf {x}}_{k}|{\textbf {x}}_{k-1},{\textbf {x}}_{k-2},\dots ,{\textbf {x}}_{0})=p({\textbf {x}}_{k}|{\textbf {x}}_{k-1})}</annotation>

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</math></span><img src="./804920d5536549aed9f93171dffa6c585fed2b37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:37.916ex; height:2.843ex;" alt="{\displaystyle p({\textbf {x}}_{k}|{\textbf {x}}_{k-1},{\textbf {x}}_{k-2},\dots ,{\textbf {x}}_{0})=p({\textbf {x}}_{k}|{\textbf {x}}_{k-1})}" loading="lazy"></span></dd></dl>

<p>Similarly, the measurement at the <i>k</i>-th timestep is dependent only upon the current state, so is conditionally independent of all other states given the current state.

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<annotation encoding="application/x-tex">{\displaystyle p({\textbf {z}}_{k}|{\textbf {x}}_{k},{\textbf {x}}_{k-1},\dots ,{\textbf {x}}_{0})=p({\textbf {z}}_{k}|{\textbf {x}}_{k})}</annotation>

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</math></span><img src="./9d8fbded717d1dbcbb8cf8f9ae434cb4fd81a6ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:33.269ex; height:2.843ex;" alt="{\displaystyle p({\textbf {z}}_{k}|{\textbf {x}}_{k},{\textbf {x}}_{k-1},\dots ,{\textbf {x}}_{0})=p({\textbf {z}}_{k}|{\textbf {x}}_{k})}" loading="lazy"></span></dd></dl>

<p>Using these assumptions the probability distribution over all states of the HMM can be written simply as

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<annotation encoding="application/x-tex">{\displaystyle p({\textbf {x}}_{0},\dots ,{\textbf {x}}_{k},{\textbf {z}}_{1},\dots ,{\textbf {z}}_{k})=p({\textbf {x}}_{0})\prod _{i=1}^{k}p({\textbf {z}}_{i}|{\textbf {x}}_{i})p({\textbf {x}}_{i}|{\textbf {x}}_{i-1}).}</annotation>

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</math></span><img src="./52d29f5877d5766e62ed7ec02317d3dcc6f48d87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:54.847ex; height:7.343ex;" alt="{\displaystyle p({\textbf {x}}_{0},\dots ,{\textbf {x}}_{k},{\textbf {z}}_{1},\dots ,{\textbf {z}}_{k})=p({\textbf {x}}_{0})\prod _{i=1}^{k}p({\textbf {z}}_{i}|{\textbf {x}}_{i})p({\textbf {x}}_{i}|{\textbf {x}}_{i-1}).}" loading="lazy"></span></dd></dl>

<p>However, when using the Kalman filter to estimate the state <b>x</b>, the probability distribution of interest is associated with the current states conditioned on the measurements up to the current timestep. (This is achieved by marginalising out the previous states and dividing by the probability of the measurement set.)

</p><p>This leads to the <i>predict</i> and <i>update</i> steps of the Kalman filter written probabilistically. The probability distribution associated with the predicted state is the sum (integral) of the products of the probability distribution associated with the transition from the (<i>k</i> - 1)-th timestep to the <i>k</i>-th and the probability distribution associated with the previous state, over all possible <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k-1}}">

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

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</math></span><img src="./9c60c4443107198983b1ced988c34b238bcd9119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.519ex; height:2.009ex;" alt="{\displaystyle x_{k-1}}" loading="lazy"></span>.

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<annotation encoding="application/x-tex">{\displaystyle p({\textbf {x}}_{k}|{\textbf {z}}_{1:k-1})=\int p({\textbf {x}}_{k}|{\textbf {x}}_{k-1})p({\textbf {x}}_{k-1}|{\textbf {z}}_{1:k-1})\,d{\textbf {x}}_{k-1}}</annotation>

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</math></span><img src="./e8212367f2ce094f326d9e121b4cc1e703036bdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:48.362ex; height:5.676ex;" alt="{\displaystyle p({\textbf {x}}_{k}|{\textbf {z}}_{1:k-1})=\int p({\textbf {x}}_{k}|{\textbf {x}}_{k-1})p({\textbf {x}}_{k-1}|{\textbf {z}}_{1:k-1})\,d{\textbf {x}}_{k-1}}" loading="lazy"></span></dd></dl>

<p>The probability distribution of update is proportional to the product of the measurement likelihood and the predicted state.

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<annotation encoding="application/x-tex">{\displaystyle p({\textbf {x}}_{k}|{\textbf {z}}_{1:k})={\frac {p({\textbf {z}}_{k}|{\textbf {x}}_{k})p({\textbf {x}}_{k}|{\textbf {z}}_{1:k-1})}{p({\textbf {z}}_{k}|{\textbf {z}}_{1:k-1})}}\propto p({\textbf {z}}_{k}|{\textbf {x}}_{k})p({\textbf {x}}_{k}|{\textbf {z}}_{1:k-1})}</annotation>

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</math></span><img src="./c83394f341515453ec80d03c368eb134d6822740.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:57.172ex; height:6.509ex;" alt="{\displaystyle p({\textbf {x}}_{k}|{\textbf {z}}_{1:k})={\frac {p({\textbf {z}}_{k}|{\textbf {x}}_{k})p({\textbf {x}}_{k}|{\textbf {z}}_{1:k-1})}{p({\textbf {z}}_{k}|{\textbf {z}}_{1:k-1})}}\propto p({\textbf {z}}_{k}|{\textbf {x}}_{k})p({\textbf {x}}_{k}|{\textbf {z}}_{1:k-1})}" loading="lazy"></span></dd></dl>

<p>The denominator

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<annotation encoding="application/x-tex">{\displaystyle p({\textbf {z}}_{k}|{\textbf {z}}_{1:k-1})=\int p({\textbf {z}}_{k}|{\textbf {x}}_{k})p({\textbf {x}}_{k}|{\textbf {z}}_{1:k-1})d{\textbf {x}}_{k}}</annotation>

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</math></span><img src="./a031eea4d66d312bafdb8e3e6287c1d7469b7462.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:41.227ex; height:5.676ex;" alt="{\displaystyle p({\textbf {z}}_{k}|{\textbf {z}}_{1:k-1})=\int p({\textbf {z}}_{k}|{\textbf {x}}_{k})p({\textbf {x}}_{k}|{\textbf {z}}_{1:k-1})d{\textbf {x}}_{k}}" loading="lazy"></span></dd></dl>

<p>is constant relative to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

<ul><li><a href="Kalman_filter" title="Kalman filter">Kalman filter</a>, a recursive Bayesian filter for <a href="Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distributions</a></li>

<li><a href="Particle_filter" title="Particle filter">Particle filter</a>, a sequential Monte Carlo (SMC) based technique, which models the <a href="Probability_density_function" title="Probability density function">PDF</a> using a set of discrete points</li>

<li><b>Grid-based estimators</b>, which subdivide the PDF into a deterministic discrete grid</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Sequential_Bayesian_filtering">Sequential Bayesian filtering</h2></div>

<p>Sequential Bayesian filtering is the extension of the Bayesian estimation for the case when the observed value changes in time. It is a method to estimate the real value of an observed variable that evolves in time.

</p><p>There are several variations:

</p>

<dl><dt>filtering</dt>

<dd>when estimating the <i>current</i> value given past and current observations,</dd>

<dt><a href="Smoothing_problem" class="mw-redirect" title="Smoothing problem">smoothing</a></dt>

<dd>when estimating <i>past</i> values given past and current observations, and</dd>

<dt>prediction</dt>

<dd>when estimating a probable <i>future</i> value given past and current observations.</dd></dl>

<p>The notion of Sequential Bayesian filtering is extensively used in <a href="Control_theory" title="Control theory">control</a> and <a href="Robotics" title="Robotics">robotics</a>.

</p>

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

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</style><cite id="CITEREFArulampalamMaskellGordon2002" class="citation journal cs1">Arulampalam, M. Sanjeev; Maskell, Simon; Gordon, Neil (2002). "A Tutorial on Particle Filters for On-line Non-linear/Non-Gaussian Bayesian Tracking". <i>IEEE Transactions on Signal Processing</i>. <b>50</b> (2): <span class="nowrap">174–</span>188. <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/2002ITSP...50..174A">2002ITSP...50..174A</a>. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.117.1144">10.1.1.117.1144</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F78.978374">10.1109/78.978374</a>.</cite></li>

<li><cite id="CITEREFBurkhart2019" class="citation book cs1">Burkhart, Michael C. (2019). "Chapter 1. An Overview of Bayesian Filtering". <i>A Discriminative Approach to Bayesian Filtering with Applications to Human Neural Decoding</i>. Providence, RI, USA: Brown University. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.26300%2Fnhfp-xv22">10.26300/nhfp-xv22</a>.</cite></li>

<li><cite id="CITEREFChen2003" class="citation journal cs1">Chen, Zhe Sage (2003). "Bayesian Filtering: From Kalman Filters to Particle Filters, and Beyond". <i>Statistics: A Journal of Theoretical and Applied Statistics</i>. <b>182</b> (1): <span class="nowrap">1–</span>69.</cite></li>

<li><cite id="CITEREFDiardBessièreMazer2003" class="citation web cs1">Diard, Julien; Bessière, Pierre; Mazer, Emmanuel (2003). <a rel="nofollow" class="external text" href="http://cogprints.org/3755/1/Diard03a.pdf">"A survey of probabilistic models, using the Bayesian Programming methodology as a unifying framework"</a> <span class="cs1-format">(PDF)</span>. cogprints.org.</cite></li>

<li><cite id="CITEREFSärkkä2013" class="citation book cs1">Särkkä, Simo (2013). <a rel="nofollow" class="external text" href="https://users.aalto.fi/~ssarkka/pub/cup_book_online_20131111.pdf"><i>Bayesian Filtering and Smoothing</i></a> <span class="cs1-format">(PDF)</span>. Cambridge University Press.</cite></li>

<li><cite id="CITEREFVolkov2015" class="citation journal cs1">Volkov, Alexander (2015). "Accuracy bounds of non-Gaussian Bayesian tracking in a NLOS environment". <i>Signal Processing</i>. <b>108</b>: <span class="nowrap">498–</span>508. <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/2015SigPr.108..498V">2015SigPr.108..498V</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.sigpro.2014.10.025">10.1016/j.sigpro.2014.10.025</a>.</cite></li></ul></div><div><div class="zim-footer">

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