Dirichlet process - Wikipedia

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class="vector-toc-numb">2</span> <span>Formal definition</span> </div> </a> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternative_views" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Alternative_views"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Alternative views</span> </div> </a> <button aria-controls="toc-Alternative_views-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Alternative views subsection</span> </button> <ul id="toc-Alternative_views-sublist" class="vector-toc-list"> <li id="toc-The_Chinese_restaurant_process" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Chinese_restaurant_process"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>The Chinese restaurant process</span> </div> </a> <ul id="toc-The_Chinese_restaurant_process-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_stick-breaking_process" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_stick-breaking_process"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>The stick-breaking process</span> </div> </a> <ul id="toc-The_stick-breaking_process-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Pólya_urn_scheme" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Pólya_urn_scheme"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>The Pólya urn scheme</span> </div> </a> <ul id="toc-The_Pólya_urn_scheme-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Use_as_a_prior_distribution" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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<span class="vector-toc-numb">4.2</span> <span>Posterior consistency</span> </div> </a> <ul id="toc-Posterior_consistency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bernstein–Von_Mises_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bernstein–Von_Mises_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Bernstein–Von Mises theorem</span> </div> </a> <ul id="toc-Bernstein–Von_Mises_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Use_in_Dirichlet_mixture_models" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Use_in_Dirichlet_mixture_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Use in Dirichlet mixture models</span> </div> </a> <button aria-controls="toc-Use_in_Dirichlet_mixture_models-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only 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href="#Applications_of_the_Dirichlet_process"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Applications of the Dirichlet process</span> </div> </a> <ul id="toc-Applications_of_the_Dirichlet_process-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_distributions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Related distributions</span> </div> </a> <ul id="toc-Related_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Dirichlet_process_draws.svg/450px-Dirichlet_process_draws.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Dirichlet_process_draws.svg/600px-Dirichlet_process_draws.svg.png 2x" data-file-width="900" data-file-height="900" /></a><figcaption>Draws from the Dirichlet process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {DP} (N(0,1),\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>DP</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {DP} (N(0,1),\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49c42590777fda40e1f471eea2571c4e03796b52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.921ex; height:2.843ex;" alt="{\displaystyle \operatorname {DP} (N(0,1),\alpha )}"></span>. The four rows use different alpha <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> (top to bottom: 1, 10, 100 and 1000) and each row contains three repetitions of the same experiment. As seen from the graphs, draws from a Dirichlet process are discrete distributions and they become less concentrated (more spread out) with increasing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. The graphs were generated using the <a href="/wiki/Stick-breaking_process" class="mw-redirect" title="Stick-breaking process">stick-breaking process</a> view of the Dirichlet process.</figcaption></figure> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, <b>Dirichlet processes</b> (after the distribution associated with <a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a>) are a family of <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic processes</a> whose <a href="/wiki/Realization_(probability)" title="Realization (probability)">realizations</a> are <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a>. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a> to describe the <a href="/wiki/Prior_probability" title="Prior probability">prior</a> knowledge about the distribution of <a href="/wiki/Random_variable" title="Random variable">random variables</a>—how likely it is that the random variables are distributed according to one or another particular distribution. </p><p>As an example, a bag of 100 real-world dice is a <i>random <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a> (random pmf)</i>—to sample this random pmf you put your hand in the bag and draw out a die, that is, you draw a pmf. A bag of dice manufactured using a crude process 100 years ago will likely have probabilities that deviate wildly from the uniform pmf, whereas a bag of state-of-the-art dice used by Las Vegas casinos may have barely perceptible imperfections. We can model the randomness of pmfs with the Dirichlet distribution.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The Dirichlet process is specified by a base distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" 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="{\displaystyle H}"></span> and a positive <a href="/wiki/Real_number" title="Real number">real number</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> called the <a href="/wiki/Concentration_parameter" title="Concentration parameter">concentration parameter</a> (also known as scaling parameter). The base distribution is the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of the process, i.e., the Dirichlet process draws distributions "around" the base distribution the way a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> draws real numbers around its mean. However, even if the base distribution is <a href="/wiki/Continuous_distribution" class="mw-redirect" title="Continuous distribution">continuous</a>, the distributions drawn from the Dirichlet process are <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a> <a href="/wiki/Discrete_distribution" class="mw-redirect" title="Discrete distribution">discrete</a>. The scaling parameter specifies how strong this discretization is: in the limit of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \rightarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2b1de74c597df1a88852a36f8ed46e3af14fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.264ex; height:2.176ex;" alt="{\displaystyle \alpha \rightarrow 0}"></span>, the realizations are all concentrated at a single value, while in the limit of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0337f6405bed43834d138cd765f090040bc81870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.425ex; height:1.843ex;" alt="{\displaystyle \alpha \rightarrow \infty }"></span> the realizations become continuous. Between the two extremes the realizations are discrete distributions with less and less concentration as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> increases. </p><p>The Dirichlet process can also be seen as the infinite-dimensional generalization of the <a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet distribution</a>. In the same way as the Dirichlet distribution is the <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> for the <a href="/wiki/Categorical_distribution" title="Categorical distribution">categorical distribution</a>, the Dirichlet process is the conjugate prior for infinite, <a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">nonparametric</a> discrete distributions. A particularly important application of Dirichlet processes is as a <a href="/wiki/Prior_probability" title="Prior probability">prior probability</a> distribution in <a href="/wiki/Infinite_mixture_model" class="mw-redirect" title="Infinite mixture model">infinite mixture models</a>. </p><p>The Dirichlet process was formally introduced by <a href="/wiki/Thomas_S._Ferguson" title="Thomas S. Ferguson">Thomas S. Ferguson</a> in 1973.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It has since been applied in <a href="/wiki/Data_mining" title="Data mining">data mining</a> and <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, among others for <a href="/wiki/Natural_language_processing" title="Natural language processing">natural language processing</a>, <a href="/wiki/Computer_vision" title="Computer vision">computer vision</a> and <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dirichlet processes are usually used when modelling data that tends to repeat previous values in a so-called "rich get richer" fashion. Specifically, suppose that the generation of values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d284b1a1ab85cfbe91d035c6a3bc7a1d17cd31a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.748ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\dots }"></span> can be simulated by the following algorithm. </p> <dl><dd><b>Input:</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" 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="{\displaystyle H}"></span> (a probability distribution called base distribution), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> (a positive real number called <a href="/wiki/Scale_parameter" title="Scale parameter">scaling parameter</a>)</dd> <dd>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></span>:</dd></dl> <div class="poem"> <p><span class="mw-poem-indented" style="display: inline-block; margin-inline-start: 4em;"> a) With probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha }{\alpha +n-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>α</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{\alpha +n-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4aa2409e344a27e1b97881d981d1a245dff5f43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.562ex; height:4.843ex;" alt="{\displaystyle {\frac {\alpha }{\alpha +n-1}}}"></span> draw <span class="mwe-math-element"><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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" 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="{\displaystyle H}"></span>.</span><br /> <br /> <span class="mw-poem-indented" style="display: inline-block; margin-inline-start: 4em;"> b) With probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n_{x}}{\alpha +n-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow> <mi>α</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n_{x}}{\alpha +n-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7c85243a7061539bce80cc56f842514e4c5025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.562ex; height:4.843ex;" alt="{\displaystyle {\frac {n_{x}}{\alpha +n-1}}}"></span> set <span class="mwe-math-element"><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_{n}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/287af9a274b87a77a0c69669ae54fc8e032c9a28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.571ex; height:2.509ex;" alt="{\displaystyle X_{n}=x}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a811d9f1857b131167bd4238be8eb175ae3e85d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.009ex;" alt="{\displaystyle n_{x}}"></span> is the number of previous observations of <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"></span>.</span><br /> <span class="mw-poem-indented" style="display: inline-block; margin-inline-start: 5em;"> (Formally, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{x}:=|\{j\colon X_{j}=x{\text{ and }}j<n\}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>j</mi> <mo>:</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>j</mi> <mo><</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{x}:=|\{j\colon X_{j}=x{\text{ and }}j<n\}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2d4bba6c3cdb4bd5ca426e8a761382be4b8f58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.545ex; height:3.009ex;" alt="{\displaystyle n_{x}:=|\{j\colon X_{j}=x{\text{ and }}j<n\}|}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cdot |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4570d0a1c9fb8f2f413f0b73ce846dd1eb1dca3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.973ex; height:2.843ex;" alt="{\displaystyle |\cdot |}"></span> denotes the number of elements in the set.) </span> </p> </div> <p>At the same time, another common model for data is that the observations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d284b1a1ab85cfbe91d035c6a3bc7a1d17cd31a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.748ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\dots }"></span> are assumed to be <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed</a> (i.i.d.) according to some (random) distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. The goal of introducing Dirichlet processes is to be able to describe the procedure outlined above in this i.i.d. model. </p><p>The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d284b1a1ab85cfbe91d035c6a3bc7a1d17cd31a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.748ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\dots }"></span> observations in the algorithm are not <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independent</a>, since we have to consider the previous results when generating the next value. They are, however, <a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">exchangeable</a>. This fact can be shown by calculating the <a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">joint probability distribution</a> of the observations and noticing that the resulting formula only depends on which <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"></span> values occur among the observations and how many repetitions they each have. Because of this exchangeability, <a href="/wiki/De_Finetti%27s_theorem" title="De Finetti's theorem">de Finetti's representation theorem</a> applies and it implies that the observations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d284b1a1ab85cfbe91d035c6a3bc7a1d17cd31a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.748ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\dots }"></span> are <a href="/wiki/Conditional_independence" title="Conditional independence">conditionally independent</a> given a (latent) distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. This <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is a random variable itself and has a distribution. This distribution (over distributions) is called a Dirichlet process (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {DP} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>DP</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {DP} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86df9b0ed157ed9e5ace00e8cf4711af36405c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.358ex; height:2.176ex;" alt="{\displaystyle \operatorname {DP} }"></span>). In summary, this means that we get an equivalent procedure to the above algorithm: </p> <ol><li>Draw a distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {DP} \left(H,\alpha \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>DP</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>H</mi> <mo>,</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {DP} \left(H,\alpha \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40cec10e31c55e0815d7fb2bef34e285cecd3dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.753ex; height:2.843ex;" alt="{\displaystyle \operatorname {DP} \left(H,\alpha \right)}"></span></li> <li>Draw observations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d284b1a1ab85cfbe91d035c6a3bc7a1d17cd31a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.748ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\dots }"></span> independently from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>.</li></ol> <p>In practice, however, drawing a concrete distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is impossible, since its specification requires an infinite amount of information. This is a common phenomenon in the context of Bayesian <a href="/wiki/Non-parametric_statistics" class="mw-redirect" title="Non-parametric statistics">non-parametric statistics</a> where a typical task is to learn distributions on function spaces, which involve effectively infinitely many parameters. The key insight is that in many applications the infinite-dimensional distributions appear only as an intermediary computational device and are not required for either the initial specification of prior beliefs or for the statement of the final inference. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=2" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a <a href="/wiki/Measurable_set" class="mw-redirect" title="Measurable set">measurable set</a> <i>S</i>, a base probability distribution <i>H</i> and a positive <a href="/wiki/Real_number" title="Real number">real number</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, the Dirichlet process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {DP} (H,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>DP</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {DP} (H,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b584372659482cf14829397bcc0c3d25d6950d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.753ex; height:2.843ex;" alt="{\displaystyle \operatorname {DP} (H,\alpha )}"></span> is a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> whose <a href="/w/index.php?title=Sample_path&action=edit&redlink=1" class="new" title="Sample path (page does not exist)">sample path</a> (or <a href="/wiki/Realization_(probability)" title="Realization (probability)">realization</a>, i.e. an infinite sequence of <a href="/wiki/Random_variate" title="Random variate">random variates</a> drawn from the process) is a probability distribution over <i>S</i>, such that the following holds. For any measurable finite <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of <i>S</i>, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{B_{i}\}_{i=1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{B_{i}\}_{i=1}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d658daa5075d18ac0ff43449ced69b902cff6908" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.789ex; height:3.009ex;" alt="{\displaystyle \{B_{i}\}_{i=1}^{n}}"></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{if }}X\sim \operatorname {DP} (H,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>X</mi> <mo>∼</mo> <mi>DP</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{if }}X\sim \operatorname {DP} (H,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/665a2001fdb57106996d82b1d020e8f6110a43cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.771ex; height:2.843ex;" alt="{\displaystyle {\text{if }}X\sim \operatorname {DP} (H,\alpha )}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{then }}(X(B_{1}),\dots ,X(B_{n}))\sim \operatorname {Dir} (\alpha H(B_{1}),\dots ,\alpha H(B_{n})),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>then </mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>X</mi> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∼</mo> <mi>Dir</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>α</mi> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{then }}(X(B_{1}),\dots ,X(B_{n}))\sim \operatorname {Dir} (\alpha H(B_{1}),\dots ,\alpha H(B_{n})),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f93adbd39ca62a728c796bfd9955af3748d7515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.058ex; height:2.843ex;" alt="{\displaystyle {\text{then }}(X(B_{1}),\dots ,X(B_{n}))\sim \operatorname {Dir} (\alpha H(B_{1}),\dots ,\alpha H(B_{n})),}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Dir} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Dir</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Dir} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e172ae4ee5780b2e493f19d57098f518c4e342f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.334ex; height:2.176ex;" alt="{\displaystyle \operatorname {Dir} }"></span> denotes the <a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet distribution</a> and the notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe70fe22f86b76b7676dd83997623846c2c0cbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.003ex; height:2.176ex;" alt="{\displaystyle X\sim D}"></span> means that the random variable <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> has the distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Alternative_views">Alternative views</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=3" title="Edit section: Alternative views"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are several equivalent views of the Dirichlet process. Besides the formal definition above, the Dirichlet process can be defined implicitly through de Finetti's theorem as described in the first section; this is often called the <a href="/wiki/Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a>. A third alternative is the <a href="/wiki/Stick-breaking_process" class="mw-redirect" title="Stick-breaking process">stick-breaking process</a>, which defines the Dirichlet process constructively by writing a distribution sampled from the process as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{k=1}^{\infty }\beta _{k}\delta _{x_{k}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{k=1}^{\infty }\beta _{k}\delta _{x_{k}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3b05f48710761a0661d8d3fe6d9f7c55a17880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.866ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{k=1}^{\infty }\beta _{k}\delta _{x_{k}}(x)}"></span>, where <span class="mwe-math-element"><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}\}_{k=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x_{k}\}_{k=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b74df051f469cae486202548cb3380b23e16bf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.933ex; height:3.009ex;" alt="{\displaystyle \{x_{k}\}_{k=1}^{\infty }}"></span> are samples from the base distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" 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="{\displaystyle H}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{x_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45ef9366a125a58578e7535103d2c08b491f13f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.064ex; height:3.009ex;" alt="{\displaystyle \delta _{x_{k}}}"></span> is an <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> centered on <span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2b88c64c76a03611549fb9b4cf4ed060b56002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}"></span> (zero everywhere except for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x_{k}}(x_{k})=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{x_{k}}(x_{k})=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1535a32abfdcc2a23cadecf798fbdbb00fa0b965" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.553ex; height:3.009ex;" alt="{\displaystyle \delta _{x_{k}}(x_{k})=1}"></span>) and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026079ab88c28912592bb6e6d2a096f8253a5fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.404ex; height:2.509ex;" alt="{\displaystyle \beta _{k}}"></span> are defined by a recursive scheme that repeatedly samples from the <a href="/wiki/Beta_distribution" title="Beta distribution">beta distribution</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Beta} (1,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Beta</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Beta} (1,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47c71314c03acd3375a24663118a39c83a66605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.238ex; height:2.843ex;" alt="{\displaystyle \operatorname {Beta} (1,\alpha )}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="The_Chinese_restaurant_process">The Chinese restaurant process</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=4" title="Edit section: The Chinese restaurant process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></div> <figure typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Chinese_Restaurant_Process_for_DP%280.5%2CH%29.webm/800px--Chinese_Restaurant_Process_for_DP%280.5%2CH%29.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="800" height="276" data-durationhint="27" data-mwtitle="Chinese_Restaurant_Process_for_DP(0.5,H).webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Chinese_Restaurant_Process_for_DP(0.5,H).webm"><source src="//upload.wikimedia.org/wikipedia/commons/9/93/Chinese_Restaurant_Process_for_DP%280.5%2CH%29.webm#t=0,00:00:25.866" type="video/webm; codecs="vp9"" data-width="800" data-height="276" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/93/Chinese_Restaurant_Process_for_DP%280.5%2CH%29.webm/Chinese_Restaurant_Process_for_DP%280.5%2CH%29.webm.144p.mjpeg.mov#t=0,00:00:25.866" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="88" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/93/Chinese_Restaurant_Process_for_DP%280.5%2CH%29.webm/Chinese_Restaurant_Process_for_DP%280.5%2CH%29.webm.240p.vp9.webm#t=0,00:00:25.866" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="146" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/93/Chinese_Restaurant_Process_for_DP%280.5%2CH%29.webm/Chinese_Restaurant_Process_for_DP%280.5%2CH%29.webm.360p.vp9.webm#t=0,00:00:25.866" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="220" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/93/Chinese_Restaurant_Process_for_DP%280.5%2CH%29.webm/Chinese_Restaurant_Process_for_DP%280.5%2CH%29.webm.360p.webm#t=0,00:00:25.866" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="220" /></video></span><figcaption>Animation of a Chinese restaurant process with scaling parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1152b74ec1d0be1ceabb8d63499e80ed2d2479f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.558ex; height:2.176ex;" alt="{\displaystyle \alpha =0.5}"></span>. Tables are hidden once the customers of a table can not be displayed anymore; however, every table has infinitely many seats. (Recording of an interactive animation.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>)</figcaption></figure> <p>A widely employed metaphor for the Dirichlet process is based on the so-called <b>Chinese restaurant process</b>. The metaphor is as follows: </p><p>Imagine a Chinese restaurant in which customers enter. A new customer sits down at a table with a probability proportional to the number of customers already sitting there. Additionally, a customer opens a new table with a probability proportional to the scaling parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. After infinitely many customers entered, one obtains a probability distribution over infinitely many tables to be chosen. This probability distribution over the tables is a random sample of the probabilities of observations drawn from a Dirichlet process with scaling parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. </p><p>If one associates draws from the base measure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" 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="{\displaystyle H}"></span> with every table, the resulting distribution over the sample space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is a random sample of a Dirichlet process. The Chinese restaurant process is related to the <a href="/wiki/P%C3%B3lya_urn_model" title="Pólya urn model">Pólya urn sampling scheme</a> which yields samples from finite Dirichlet distributions. </p><p>Because customers sit at a table with a probability proportional to the number of customers already sitting at the table, two properties of the DP can be deduced: </p> <ol><li>The Dirichlet process exhibits a self-reinforcing property: The more often a given value has been sampled in the past, the more likely it is to be sampled again.</li> <li>Even if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" 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="{\displaystyle H}"></span> is a distribution over an <a href="/wiki/Uncountable_set" title="Uncountable set">uncountable set</a>, there is a nonzero probability that two samples will have exactly the same value because the probability mass will concentrate on a small number of tables.</li></ol> <div class="mw-heading mw-heading3"><h3 id="The_stick-breaking_process">The stick-breaking process</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=5" title="Edit section: The stick-breaking process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A third approach to the Dirichlet process is the so-called stick-breaking process view. Conceptually, this involves repeatedly breaking off and discarding a random fraction (sampled from a Beta distribution) of a "stick" that is initially of length 1. Remember that draws from a Dirichlet process are distributions over a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. As noted previously, the distribution drawn is discrete with probability 1. In the stick-breaking process view, we explicitly use the discreteness and give the <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a> of this (random) discrete distribution as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta )=\sum _{k=1}^{\infty }\beta _{k}\cdot \delta _{\theta _{k}}(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\theta )=\sum _{k=1}^{\infty }\beta _{k}\cdot \delta _{\theta _{k}}(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9a8760e24ed41d3f57a758ea943b5ae09afb5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.897ex; height:6.843ex;" alt="{\displaystyle f(\theta )=\sum _{k=1}^{\infty }\beta _{k}\cdot \delta _{\theta _{k}}(\theta )}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\theta _{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\theta _{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53dd5ace7670b08458d322f25a53a30809140253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.895ex; height:3.009ex;" alt="{\displaystyle \delta _{\theta _{k}}}"></span> is the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> which evaluates to zero everywhere, except for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\theta _{k}}(\theta _{k})=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\theta _{k}}(\theta _{k})=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be25689fb4eeecd272c8dcdacceaff3555c48c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.145ex; height:3.009ex;" alt="{\displaystyle \delta _{\theta _{k}}(\theta _{k})=1}"></span>. Since this distribution is random itself, its mass function is parameterized by two sets of random variables: the locations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\theta _{k}\right\}_{k=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>{</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\theta _{k}\right\}_{k=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41a26a21424e12b9dbe671f50a5733c771fa9f0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.693ex; height:3.176ex;" alt="{\displaystyle \left\{\theta _{k}\right\}_{k=1}^{\infty }}"></span> and the corresponding probabilities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\beta _{k}\right\}_{k=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>{</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\beta _{k}\right\}_{k=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cedfd046835dab17fe8b72a4337a4a84576838b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.919ex; height:3.176ex;" alt="{\displaystyle \left\{\beta _{k}\right\}_{k=1}^{\infty }}"></span>. In the following, we present without proof what these random variables are. </p><p>The locations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae772f574aa637900f7673f6f934bdcf35969ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.179ex; height:2.509ex;" alt="{\displaystyle \theta _{k}}"></span> are independent and identically distributed according to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" 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="{\displaystyle H}"></span>, the base distribution of the Dirichlet process. The probabilities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026079ab88c28912592bb6e6d2a096f8253a5fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.404ex; height:2.509ex;" alt="{\displaystyle \beta _{k}}"></span> are given by a procedure resembling the breaking of a unit-length stick (hence the name): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{k}=\beta '_{k}\cdot \prod _{i=1}^{k-1}\left(1-\beta '_{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> <mo>⋅</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{k}=\beta '_{k}\cdot \prod _{i=1}^{k-1}\left(1-\beta '_{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce305f895718f6439d40a634ff120da15ae46c25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.191ex; height:7.343ex;" alt="{\displaystyle \beta _{k}=\beta '_{k}\cdot \prod _{i=1}^{k-1}\left(1-\beta '_{i}\right)}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta '_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff03438ebb1fe546e70391802f010cc2e0ca7d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.404ex; height:2.843ex;" alt="{\displaystyle \beta '_{k}}"></span> are independent random variables with the <a href="/wiki/Beta_distribution" title="Beta distribution">beta distribution</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Beta} (1,\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Beta</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Beta} (1,\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47c71314c03acd3375a24663118a39c83a66605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.238ex; height:2.843ex;" alt="{\displaystyle \operatorname {Beta} (1,\alpha )}"></span>. The resemblance to 'stick-breaking' can be seen by considering <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026079ab88c28912592bb6e6d2a096f8253a5fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.404ex; height:2.509ex;" alt="{\displaystyle \beta _{k}}"></span> as the length of a piece of a stick. We start with a unit-length stick and in each step we break off a portion of the remaining stick according to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta '_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff03438ebb1fe546e70391802f010cc2e0ca7d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.404ex; height:2.843ex;" alt="{\displaystyle \beta '_{k}}"></span> and assign this broken-off piece to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026079ab88c28912592bb6e6d2a096f8253a5fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.404ex; height:2.509ex;" alt="{\displaystyle \beta _{k}}"></span>. The formula can be understood by noting that after the first <i>k</i> − 1 values have their portions assigned, the length of the remainder of the stick is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{i=1}^{k-1}\left(1-\beta '_{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{i=1}^{k-1}\left(1-\beta '_{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31bd51783edcfe9783f3e253845c90101eca972c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.605ex; height:7.343ex;" alt="{\displaystyle \prod _{i=1}^{k-1}\left(1-\beta '_{i}\right)}"></span> and this piece is broken according to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta '_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta '_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff03438ebb1fe546e70391802f010cc2e0ca7d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.404ex; height:2.843ex;" alt="{\displaystyle \beta '_{k}}"></span> and gets assigned to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026079ab88c28912592bb6e6d2a096f8253a5fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.404ex; height:2.509ex;" alt="{\displaystyle \beta _{k}}"></span>. </p><p>The smaller <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is, the less of the stick will be left for subsequent values (on average), yielding more concentrated distributions. </p><p>The stick-breaking process is similar to the construction where one samples sequentially from <a href="/wiki/Dirichlet_distribution#From_marginal_beta_distributions" title="Dirichlet distribution">marginal beta distributions</a> in order to generate a sample from a <a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet distribution</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="The_Pólya_urn_scheme"><span id="The_P.C3.B3lya_urn_scheme"></span>The Pólya urn scheme</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=6" title="Edit section: The Pólya urn scheme"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Yet another way to visualize the Dirichlet process and Chinese restaurant process is as a modified <a href="/wiki/P%C3%B3lya_urn_model" title="Pólya urn model">Pólya urn scheme</a> sometimes called the <b>Blackwell–MacQueen</b> sampling scheme. Imagine that we start with an urn filled with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> black balls. Then we proceed as follows: </p> <ol><li>Each time we need an observation, we draw a ball from the urn.</li> <li>If the ball is black, we generate a new (non-black) colour uniformly, label a new ball this colour, drop the new ball into the urn along with the ball we drew, and return the colour we generated.</li> <li>Otherwise, label a new ball with the colour of the ball we drew, drop the new ball into the urn along with the ball we drew, and return the colour we observed.</li></ol> <p>The resulting distribution over colours is the same as the distribution over tables in the Chinese restaurant process. Furthermore, when we draw a black ball, if rather than generating a new colour, we instead pick a random value from a base distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" 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="{\displaystyle H}"></span> and use that value to label the new ball, the resulting distribution over labels will be the same as the distribution over the values in a Dirichlet process. </p> <div class="mw-heading mw-heading2"><h2 id="Use_as_a_prior_distribution">Use as a prior distribution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=7" title="Edit section: Use as a prior distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Dirichlet Process can be used as a prior distribution to estimate the probability distribution that generates the data. In this section, we consider the model </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P&\sim {\textrm {DP}}(H,\alpha )\\X_{1},\ldots ,X_{n}\mid P&\,{\overset {\textrm {i.i.d.}}{\sim }}\,P.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>P</mi> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>DP</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <mi>P</mi> </mtd> <mtd> <mi></mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>i.i.d.</mtext> </mrow> </mover> </mrow> <mspace width="thinmathspace" /> <mi>P</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P&\sim {\textrm {DP}}(H,\alpha )\\X_{1},\ldots ,X_{n}\mid P&\,{\overset {\textrm {i.i.d.}}{\sim }}\,P.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cab2dfd6241ac2bd97814471d003da8815dd25c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.585ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}P&\sim {\textrm {DP}}(H,\alpha )\\X_{1},\ldots ,X_{n}\mid P&\,{\overset {\textrm {i.i.d.}}{\sim }}\,P.\end{aligned}}}"></span></dd></dl> <p>The Dirichlet Process distribution satisfies <a href="/wiki/Conjugate_prior" title="Conjugate prior">prior conjugacy</a>, posterior consistency, and the <a href="/wiki/Bernstein%E2%80%93von_Mises_theorem" title="Bernstein–von Mises theorem">Bernstein–von Mises theorem</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Prior_conjugacy">Prior conjugacy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=8" title="Edit section: Prior conjugacy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this model, the posterior distribution is again a Dirichlet process. This means that the Dirichlet process is a <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> for this model. The <a href="/wiki/Posterior_distribution" class="mw-redirect" title="Posterior distribution">posterior distribution</a> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P\mid X_{1},\ldots ,X_{n}&\sim {\textrm {DP}}\left({\frac {\alpha }{\alpha +n}}H+{\frac {1}{\alpha +n}}\sum _{i=1}^{n}\delta _{X_{i}},\;\alpha +n\right)\\&={\textrm {DP}}\left({\frac {\alpha }{\alpha +n}}H+{\frac {n}{\alpha +n}}\mathbb {P} _{n},\;\alpha +n\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>P</mi> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>DP</mtext> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>α</mi> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mi>H</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>α</mi> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mi>α</mi> <mo>+</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>DP</mtext> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>α</mi> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mi>H</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>α</mi> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mi>α</mi> <mo>+</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P\mid X_{1},\ldots ,X_{n}&\sim {\textrm {DP}}\left({\frac {\alpha }{\alpha +n}}H+{\frac {1}{\alpha +n}}\sum _{i=1}^{n}\delta _{X_{i}},\;\alpha +n\right)\\&={\textrm {DP}}\left({\frac {\alpha }{\alpha +n}}H+{\frac {n}{\alpha +n}}\mathbb {P} _{n},\;\alpha +n\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38f65dbb759fe97dc345a22bd1559a382315e20e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:59.062ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}P\mid X_{1},\ldots ,X_{n}&\sim {\textrm {DP}}\left({\frac {\alpha }{\alpha +n}}H+{\frac {1}{\alpha +n}}\sum _{i=1}^{n}\delta _{X_{i}},\;\alpha +n\right)\\&={\textrm {DP}}\left({\frac {\alpha }{\alpha +n}}H+{\frac {n}{\alpha +n}}\mathbb {P} _{n},\;\alpha +n\right)\end{aligned}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f80c6ddb8e46a9f50e556e21219ff23fbbb696cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.639ex; height:2.509ex;" alt="{\displaystyle \mathbb {P} _{n}}"></span> is defined below. </p> <div class="mw-heading mw-heading3"><h3 id="Posterior_consistency">Posterior consistency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=9" title="Edit section: Posterior consistency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we take the <a href="/wiki/Frequentist" class="mw-redirect" title="Frequentist">frequentist</a> view of probability, we believe there is a true probability distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"></span> that generated the data. Then it turns out that the Dirichlet process is consistent in the <a href="/wiki/Weak_topology" title="Weak topology">weak topology</a>, which means that for every weak neighbourhood <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"></span>, the posterior probability of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> converges to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Bernstein–Von_Mises_theorem"><span id="Bernstein.E2.80.93Von_Mises_theorem"></span>Bernstein–Von Mises theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=10" title="Edit section: Bernstein–Von Mises theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In order to interpret the credible sets as confidence sets, a <a href="/wiki/Bernstein%E2%80%93von_Mises_theorem" title="Bernstein–von Mises theorem">Bernstein–von Mises theorem</a> is needed. In case of the Dirichlet process we compare the posterior distribution with the <a href="/wiki/Empirical_process" title="Empirical process">empirical process</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} _{n}={\frac {1}{n}}\sum _{i=1}^{n}\delta _{X_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} _{n}={\frac {1}{n}}\sum _{i=1}^{n}\delta _{X_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4626d122b3e87bf374a999702c4571d4b724375c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.347ex; height:6.843ex;" alt="{\displaystyle \mathbb {P} _{n}={\frac {1}{n}}\sum _{i=1}^{n}\delta _{X_{i}}}"></span>. Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"></span>-Donsker class, i.e. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}\left(\mathbb {P} _{n}-P_{0}\right)\rightsquigarrow G_{P_{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">⇝</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}\left(\mathbb {P} _{n}-P_{0}\right)\rightsquigarrow G_{P_{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62d7abc5376cad2ed0d980b883ad95abea019383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.112ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}\left(\mathbb {P} _{n}-P_{0}\right)\rightsquigarrow G_{P_{0}}}"></span></dd></dl> <p>for some Brownian Bridge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{P_{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{P_{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb7e2ca86896a0d5a25e5f6b8623525053ec06f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.946ex; height:2.843ex;" alt="{\displaystyle G_{P_{0}}}"></span>. Suppose also that there exists a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)\geq \sup _{f\in {\mathcal {F}}}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)\geq \sup _{f\in {\mathcal {F}}}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e378a2c559c85e038d183cffc87b6faab5606b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.284ex; height:4.843ex;" alt="{\displaystyle F(x)\geq \sup _{f\in {\mathcal {F}}}f(x)}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int F^{2}\,\mathrm {d} H<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int F^{2}\,\mathrm {d} H<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6498a8dff03af9508f132dd427787b4e6a9d99d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.615ex; height:5.676ex;" alt="{\displaystyle \int F^{2}\,\mathrm {d} H<\infty }"></span>, then, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"></span> <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}\left(P-\mathbb {P} _{n}\right)\mid X_{1},\cdots ,X_{n}\rightsquigarrow G_{P_{0}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mi>P</mi> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">⇝</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}\left(P-\mathbb {P} _{n}\right)\mid X_{1},\cdots ,X_{n}\rightsquigarrow G_{P_{0}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40840972745938b1676fdc5069e5302d1d08fdcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.195ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}\left(P-\mathbb {P} _{n}\right)\mid X_{1},\cdots ,X_{n}\rightsquigarrow G_{P_{0}}.}"></span></dd></dl> <p>This implies that credible sets you construct are asymptotic confidence sets, and the Bayesian inference based on the Dirichlet process is asymptotically also valid frequentist inference. </p> <div class="mw-heading mw-heading2"><h2 id="Use_in_Dirichlet_mixture_models">Use in Dirichlet mixture models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=11" title="Edit section: Use in Dirichlet mixture models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:DP_clustering_simulation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/DP_clustering_simulation.png/220px-DP_clustering_simulation.png" decoding="async" width="220" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/DP_clustering_simulation.png/330px-DP_clustering_simulation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/DP_clustering_simulation.png/440px-DP_clustering_simulation.png 2x" data-file-width="579" data-file-height="380" /></a><figcaption>Simulation of 1000 observations drawn from a Dirichlet mixture model. Each observation within a cluster is drawn independently from the <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\mu _{k},1/4)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\mu _{k},1/4)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84e66e7ce308ce976bfb3cc97412f1da8010ad84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.885ex; height:2.843ex;" alt="{\displaystyle N(\mu _{k},1/4)}"></span>. The cluster means <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f9c464114cfcd7e31e53de202703377b7c5ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.49ex; height:2.176ex;" alt="{\displaystyle \mu _{k}}"></span> are drawn from a distribution G which itself is drawn from a Dirichlet process with concentration parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1152b74ec1d0be1ceabb8d63499e80ed2d2479f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.558ex; height:2.176ex;" alt="{\displaystyle \alpha =0.5}"></span> and base distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=N(2,16)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>16</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=N(2,16)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85d02ad03300bff52783ade73b0e2dba8d75960f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.556ex; height:2.843ex;" alt="{\displaystyle H=N(2,16)}"></span>. Each row is a new simulation.</figcaption></figure> <p>To understand what Dirichlet processes are and the problem they solve we consider the example of <a href="/wiki/Data_clustering" class="mw-redirect" title="Data clustering">data clustering</a>. It is a common situation that data points are assumed to be distributed in a hierarchical fashion where each data point belongs to a (randomly chosen) cluster and the members of a cluster are further distributed randomly within that cluster. </p> <div class="mw-heading mw-heading3"><h3 id="Example_1">Example 1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=12" title="Edit section: Example 1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For example, we might be interested in how people will vote on a number of questions in an upcoming election. A reasonable model for this situation might be to classify each voter as a liberal, a conservative or a moderate and then model the event that a voter says "Yes" to any particular question as a <a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli random variable</a> with the probability dependent on which political cluster they belong to. By looking at how votes were cast in previous years on similar pieces of legislation one could fit a predictive model using a simple clustering algorithm such as <a href="/wiki/K-means_clustering" title="K-means clustering"><i>k</i>-means</a>. That algorithm, however, requires knowing in advance the number of clusters that generated the data. In many situations, it is not possible to determine this ahead of time, and even when we can reasonably assume a number of clusters we would still like to be able to check this assumption. For example, in the voting example above the division into liberal, conservative and moderate might not be finely tuned enough; attributes such as a religion, class or race could also be critical for modelling voter behaviour, resulting in more clusters in the model. </p> <div class="mw-heading mw-heading3"><h3 id="Example_2">Example 2</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=13" title="Edit section: Example 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As another example, we might be interested in modelling the velocities of galaxies using a simple model assuming that the velocities are clustered, for instance by assuming each velocity is distributed according to the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}\sim N(\mu _{k},\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}\sim N(\mu _{k},\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d14e21e47646e6cd7912854764020542dddacc2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.808ex; height:3.176ex;" alt="{\displaystyle v_{i}\sim N(\mu _{k},\sigma ^{2})}"></span>, where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>th observation belongs to the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" 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="{\displaystyle k}"></span>th cluster of galaxies with common expected velocity. In this case it is far from obvious how to determine a priori how many clusters (of common velocities) there should be and any model for this would be highly suspect and should be checked against the data. By using a Dirichlet process prior for the distribution of cluster means we circumvent the need to explicitly specify ahead of time how many clusters there are, although the concentration parameter still controls it implicitly. </p><p>We consider this example in more detail. A first naive model is to presuppose that there are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> clusters of normally distributed velocities with common known fixed <a href="/wiki/Variance" title="Variance">variance</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span>. Denoting the event that the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>th observation is in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" 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="{\displaystyle k}"></span>th cluster as <span class="mwe-math-element"><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_{i}=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce43ff7ad4eeb5db8cb521e50e66de8717484c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.191ex; height:2.509ex;" alt="{\displaystyle z_{i}=k}"></span> we can write this model as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(v_{i}\mid z_{i}=k,\mu _{k})&\sim N(\mu _{k},\sigma ^{2})\\\operatorname {P} (z_{i}=k)&=\pi _{k}\\({\boldsymbol {\pi }}\mid \alpha )&\sim \operatorname {Dir} \left({\frac {\alpha }{K}}\cdot \mathbf {1} _{K}\right)\\\mu _{k}&\sim H(\lambda )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">π</mi> </mrow> <mo>∣</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>Dir</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>K</mi> </mfrac> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(v_{i}\mid z_{i}=k,\mu _{k})&\sim N(\mu _{k},\sigma ^{2})\\\operatorname {P} (z_{i}=k)&=\pi _{k}\\({\boldsymbol {\pi }}\mid \alpha )&\sim \operatorname {Dir} \left({\frac {\alpha }{K}}\cdot \mathbf {1} _{K}\right)\\\mu _{k}&\sim H(\lambda )\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddce88df8985000a572a398481adf0fd5b63578c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:32.959ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}(v_{i}\mid z_{i}=k,\mu _{k})&\sim N(\mu _{k},\sigma ^{2})\\\operatorname {P} (z_{i}=k)&=\pi _{k}\\({\boldsymbol {\pi }}\mid \alpha )&\sim \operatorname {Dir} \left({\frac {\alpha }{K}}\cdot \mathbf {1} _{K}\right)\\\mu _{k}&\sim H(\lambda )\end{aligned}}}"></span></dd></dl> <p>That is, we assume that the data belongs to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> distinct clusters with means <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f9c464114cfcd7e31e53de202703377b7c5ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.49ex; height:2.176ex;" alt="{\displaystyle \mu _{k}}"></span> and that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8ff221c0357774336a9bd136e45ef96a7d31f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.414ex; height:2.009ex;" alt="{\displaystyle \pi _{k}}"></span> is the (unknown) prior probability of a data point belonging to the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" 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="{\displaystyle k}"></span>th cluster. We assume that we have no initial information distinguishing the clusters, which is captured by the symmetric prior <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Dir} \left(\alpha /K\cdot \mathbf {1} _{K}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Dir</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Dir} \left(\alpha /K\cdot \mathbf {1} _{K}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e263cebdc67db870c78203730dffc11b422f1ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.568ex; height:2.843ex;" alt="{\displaystyle \operatorname {Dir} \left(\alpha /K\cdot \mathbf {1} _{K}\right)}"></span>. Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Dir} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Dir</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Dir} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e172ae4ee5780b2e493f19d57098f518c4e342f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.334ex; height:2.176ex;" alt="{\displaystyle \operatorname {Dir} }"></span> denotes the <a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet distribution</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{K}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c653e33db07c0a2c2ea5036e69802ad6a9d659ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.03ex; height:2.509ex;" alt="{\displaystyle \mathbf {1} _{K}}"></span> denotes a vector of length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> where each element is 1. We further assign independent and identical prior distributions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(\lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d419ab1305d150c7078f0f39635c5be831dea2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.228ex; height:2.843ex;" alt="{\displaystyle H(\lambda )}"></span> to each of the cluster means, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" 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="{\displaystyle H}"></span> may be any parametric distribution with parameters denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>. The hyper-parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> are taken to be known fixed constants, chosen to reflect our prior beliefs about the system. To understand the connection to Dirichlet process priors we rewrite this model in an equivalent but more suggestive form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(v_{i}\mid {\tilde {\mu }}_{i})&\sim N({\tilde {\mu }}_{i},\sigma ^{2})\\{\tilde {\mu }}_{i}&\sim G=\sum _{k=1}^{K}\pi _{k}\delta _{\mu _{k}}({\tilde {\mu }}_{i})\\({\boldsymbol {\pi }}\mid \alpha )&\sim \operatorname {Dir} \left({\frac {\alpha }{K}}\cdot \mathbf {1} _{K}\right)\\\mu _{k}&\sim H(\lambda )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∣</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>G</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">π</mi> </mrow> <mo>∣</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>Dir</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>K</mi> </mfrac> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(v_{i}\mid {\tilde {\mu }}_{i})&\sim N({\tilde {\mu }}_{i},\sigma ^{2})\\{\tilde {\mu }}_{i}&\sim G=\sum _{k=1}^{K}\pi _{k}\delta _{\mu _{k}}({\tilde {\mu }}_{i})\\({\boldsymbol {\pi }}\mid \alpha )&\sim \operatorname {Dir} \left({\frac {\alpha }{K}}\cdot \mathbf {1} _{K}\right)\\\mu _{k}&\sim H(\lambda )\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ba8a5f15f5ce5f5760be228c4ba1614cdf5f39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:29.932ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}(v_{i}\mid {\tilde {\mu }}_{i})&\sim N({\tilde {\mu }}_{i},\sigma ^{2})\\{\tilde {\mu }}_{i}&\sim G=\sum _{k=1}^{K}\pi _{k}\delta _{\mu _{k}}({\tilde {\mu }}_{i})\\({\boldsymbol {\pi }}\mid \alpha )&\sim \operatorname {Dir} \left({\frac {\alpha }{K}}\cdot \mathbf {1} _{K}\right)\\\mu _{k}&\sim H(\lambda )\end{aligned}}}"></span></dd></dl> <p>Instead of imagining that each data point is first assigned a cluster and then drawn from the distribution associated to that cluster we now think of each observation being associated with parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\mu }}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\mu }}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7980e0e9bbbe7790d76cc3eeecd5129180507d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.676ex;" alt="{\displaystyle {\tilde {\mu }}_{i}}"></span> drawn from some discrete distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> with support on the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> means. That is, we are now treating the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\mu }}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\mu }}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7980e0e9bbbe7790d76cc3eeecd5129180507d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.676ex;" alt="{\displaystyle {\tilde {\mu }}_{i}}"></span> as being drawn from the random distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> and our prior information is incorporated into the model by the distribution over distributions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_1" poster="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Parameter_estimation_process_infinite_Gaussian_mixture_model.webm/220px--Parameter_estimation_process_infinite_Gaussian_mixture_model.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="147" data-durationhint="50" data-mwtitle="Parameter_estimation_process_infinite_Gaussian_mixture_model.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Parameter_estimation_process_infinite_Gaussian_mixture_model.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f5/Parameter_estimation_process_infinite_Gaussian_mixture_model.webm/Parameter_estimation_process_infinite_Gaussian_mixture_model.webm.360p.webm#t=0,00:00:49" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="432" data-height="288" /><source src="//upload.wikimedia.org/wikipedia/commons/f/f5/Parameter_estimation_process_infinite_Gaussian_mixture_model.webm#t=0,00:00:49" type="video/webm; codecs="vp9"" data-width="432" data-height="288" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f5/Parameter_estimation_process_infinite_Gaussian_mixture_model.webm/Parameter_estimation_process_infinite_Gaussian_mixture_model.webm.144p.mjpeg.mov#t=0,00:00:49" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="216" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f5/Parameter_estimation_process_infinite_Gaussian_mixture_model.webm/Parameter_estimation_process_infinite_Gaussian_mixture_model.webm.240p.vp9.webm#t=0,00:00:49" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="360" data-height="240" /></video></span><figcaption>Animation of the clustering process for one-dimensional data using Gaussian distributions drawn from a Dirichlet process. The histograms of the clusters are shown in different colours. During the parameter estimation process, new clusters are created and grow on the data. The legend shows the cluster colours and the number of datapoints assigned to each cluster.</figcaption></figure> <p>We would now like to extend this model to work without pre-specifying a fixed number of clusters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>. Mathematically, this means we would like to select a random prior distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G({\tilde {\mu }}_{i})=\sum _{k=1}^{\infty }\pi _{k}\delta _{\mu _{k}}({\tilde {\mu }}_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G({\tilde {\mu }}_{i})=\sum _{k=1}^{\infty }\pi _{k}\delta _{\mu _{k}}({\tilde {\mu }}_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aff1ce865275be92760a99f9601f04875a941c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.218ex; height:6.843ex;" alt="{\displaystyle G({\tilde {\mu }}_{i})=\sum _{k=1}^{\infty }\pi _{k}\delta _{\mu _{k}}({\tilde {\mu }}_{i})}"></span> where the values of the clusters means <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f9c464114cfcd7e31e53de202703377b7c5ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.49ex; height:2.176ex;" alt="{\displaystyle \mu _{k}}"></span> are again independently distributed according to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left(\lambda \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left(\lambda \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92cdf36bf7fa2060989b04ba54faf599a8f6758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.615ex; height:2.843ex;" alt="{\displaystyle H\left(\lambda \right)}"></span> and the distribution over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8ff221c0357774336a9bd136e45ef96a7d31f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.414ex; height:2.009ex;" alt="{\displaystyle \pi _{k}}"></span> is symmetric over the infinite set of clusters. This is exactly what is accomplished by the model: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(v_{i}\mid {\tilde {\mu }}_{i})&\sim N({\tilde {\mu }}_{i},\sigma ^{2})\\{\tilde {\mu }}_{i}&\sim G\\G&\sim \operatorname {DP} (H(\lambda ),\alpha )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∣</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>G</mi> </mtd> </mtr> <mtr> <mtd> <mi>G</mi> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>DP</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(v_{i}\mid {\tilde {\mu }}_{i})&\sim N({\tilde {\mu }}_{i},\sigma ^{2})\\{\tilde {\mu }}_{i}&\sim G\\G&\sim \operatorname {DP} (H(\lambda ),\alpha )\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d0d244136f4e3473ce679a00432d97381506df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:24.643ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}(v_{i}\mid {\tilde {\mu }}_{i})&\sim N({\tilde {\mu }}_{i},\sigma ^{2})\\{\tilde {\mu }}_{i}&\sim G\\G&\sim \operatorname {DP} (H(\lambda ),\alpha )\end{aligned}}}"></span></dd></dl> <p>With this in hand we can better understand the computational merits of the Dirichlet process. Suppose that we wanted to draw <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> observations from the naive model with exactly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> clusters. A simple algorithm for doing this would be to draw <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f9c464114cfcd7e31e53de202703377b7c5ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.49ex; height:2.176ex;" alt="{\displaystyle \mu _{k}}"></span> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(\lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d419ab1305d150c7078f0f39635c5be831dea2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.228ex; height:2.843ex;" alt="{\displaystyle H(\lambda )}"></span>, a distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Dir} \left(\alpha /K\cdot \mathbf {1} _{K}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Dir</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Dir} \left(\alpha /K\cdot \mathbf {1} _{K}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e263cebdc67db870c78203730dffc11b422f1ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.568ex; height:2.843ex;" alt="{\displaystyle \operatorname {Dir} \left(\alpha /K\cdot \mathbf {1} _{K}\right)}"></span> and then for each observation independently sample the cluster <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" 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="{\displaystyle k}"></span> with probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8ff221c0357774336a9bd136e45ef96a7d31f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.414ex; height:2.009ex;" alt="{\displaystyle \pi _{k}}"></span> and the value of the observation according to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\left(\mu _{k},\sigma ^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\left(\mu _{k},\sigma ^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e9b9d5bc8aa4f544c76f2500e7ea7f7949281e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.49ex; height:3.343ex;" alt="{\displaystyle N\left(\mu _{k},\sigma ^{2}\right)}"></span>. It is easy to see that this algorithm does not work in case where we allow infinite clusters because this would require sampling an infinite dimensional parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">π</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c028673bef88eb9810711b010f960ce5fe2d2e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.585ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\pi }}}"></span>. However, it is still possible to sample observations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}"></span>. One can e.g. use the Chinese restaurant representation described below and calculate the probability for used clusters and a new cluster to be created. This avoids having to explicitly specify <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">π</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c028673bef88eb9810711b010f960ce5fe2d2e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.585ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\pi }}}"></span>. Other solutions are based on a truncation of clusters: A (high) upper bound to the true number of clusters is introduced and cluster numbers higher than the lower bound are treated as one cluster. </p><p>Fitting the model described above based on observed data <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> means finding the <a href="/wiki/Posterior_distribution" class="mw-redirect" title="Posterior distribution">posterior distribution</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\left({\boldsymbol {\pi }},{\boldsymbol {\mu }}\mid D\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">π</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>∣</mo> <mi>D</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\left({\boldsymbol {\pi }},{\boldsymbol {\mu }}\mid D\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec3d1b70a7a5871aa3d36948da3f152d4779c4e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.581ex; height:2.843ex;" alt="{\displaystyle p\left({\boldsymbol {\pi }},{\boldsymbol {\mu }}\mid D\right)}"></span> over cluster probabilities and their associated means. In the infinite dimensional case it is obviously impossible to write down the posterior explicitly. It is, however, possible to draw samples from this posterior using a modified <a href="/wiki/Gibbs_sampler" class="mw-redirect" title="Gibbs sampler">Gibbs sampler</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This is the critical fact that makes the Dirichlet process prior useful for <a href="/wiki/Statistical_inference" title="Statistical inference">inference</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications_of_the_Dirichlet_process">Applications of the Dirichlet process</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=14" title="Edit section: Applications of the Dirichlet process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dirichlet processes are frequently used in <i>Bayesian <a href="/wiki/Non-parametric_statistics" class="mw-redirect" title="Non-parametric statistics">nonparametric statistics</a></i>. "Nonparametric" here does not mean a parameter-less model, rather a model in which representations grow as more data are observed. Bayesian nonparametric models have gained considerable popularity in the field of <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a> because of the above-mentioned flexibility, especially in <a href="/wiki/Unsupervised_learning" title="Unsupervised learning">unsupervised learning</a>. In a Bayesian nonparametric model, the prior and posterior distributions are not parametric distributions, but stochastic processes.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The fact that the Dirichlet distribution is a probability distribution on the <a href="/wiki/Simplex" title="Simplex">simplex</a> of sets of non-negative numbers that sum to one makes it a good candidate to model distributions over distributions or distributions over functions. Additionally, the nonparametric nature of this model makes it an ideal candidate for clustering problems where the distinct number of clusters is unknown beforehand. In addition, the Dirichlet process has also been used for developing a mixture of expert models, in the context of supervised learning algorithms (regression or classification settings). For instance, mixtures of Gaussian process experts, where the number of required experts must be inferred from the data.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>As draws from a Dirichlet process are discrete, an important use is as a <a href="/wiki/Prior_probability" title="Prior probability">prior probability</a> in <a href="/wiki/Infinite_mixture_model" class="mw-redirect" title="Infinite mixture model">infinite mixture models</a>. In this case, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is the parametric set of component distributions. The generative process is therefore that a sample is drawn from a Dirichlet process, and for each data point, in turn, a value is drawn from this sample distribution and used as the component distribution for that data point. The fact that there is no limit to the number of distinct components which may be generated makes this kind of model appropriate for the case when the number of mixture components is not well-defined in advance. For example, the infinite mixture of Gaussians model,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> as well as associated mixture regression models, e.g.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>The infinite nature of these models also lends them to <a href="/wiki/Natural_language_processing" title="Natural language processing">natural language processing</a> applications, where it is often desirable to treat the vocabulary as an infinite, discrete set. </p><p>The Dirichlet Process can also be used for nonparametric hypothesis testing, i.e. to develop Bayesian nonparametric versions of the classical nonparametric hypothesis tests, e.g. <a href="/wiki/Sign_test" title="Sign test">sign test</a>, <a href="/wiki/Wilcoxon_rank-sum_test" class="mw-redirect" title="Wilcoxon rank-sum test">Wilcoxon rank-sum test</a>, <a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Wilcoxon signed-rank test</a>, etc. For instance, Bayesian nonparametric versions of the Wilcoxon rank-sum test and the Wilcoxon signed-rank test have been developed by using the <a href="/wiki/Imprecise_Dirichlet_process" title="Imprecise Dirichlet process">imprecise Dirichlet process</a>, a prior ignorance Dirichlet process. <sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2020)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Related_distributions">Related distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=15" title="Edit section: Related distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a> is a generalization of the Dirichlet process to accommodate power-law tails</li> <li>The <a href="/wiki/Hierarchical_Dirichlet_process" title="Hierarchical Dirichlet process">hierarchical Dirichlet process</a> extends the ordinary Dirichlet process for modelling grouped data.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFrigyikKapilaGupta" class="citation web cs1">Frigyik, Bela A.; Kapila, Amol; Gupta, Maya R. <a rel="nofollow" class="external text" href="http://mayagupta.org/publications/FrigyikKapilaGuptaIntroToDirichlet.pdf">"Introduction to the Dirichlet Distribution and Related Processes"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2 September</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Introduction+to+the+Dirichlet+Distribution+and+Related+Processes&rft.aulast=Frigyik&rft.aufirst=Bela+A.&rft.au=Kapila%2C+Amol&rft.au=Gupta%2C+Maya+R.&rft_id=http%3A%2F%2Fmayagupta.org%2Fpublications%2FFrigyikKapilaGuptaIntroToDirichlet.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirichlet+process" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFerguson1973" class="citation journal cs1">Ferguson, Thomas (1973). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176342360">"Bayesian analysis of some nonparametric problems"</a>. <i><a href="/wiki/Annals_of_Statistics" title="Annals of Statistics">Annals of Statistics</a></i>. <b>1</b> (2): <span class="nowrap">209–</span>230. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176342360">10.1214/aos/1176342360</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0350949">0350949</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Statistics&rft.atitle=Bayesian+analysis+of+some+nonparametric+problems&rft.volume=1&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E209-%3C%2Fspan%3E230&rft.date=1973&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176342360&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D350949%23id-name%3DMR&rft.aulast=Ferguson&rft.aufirst=Thomas&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176342360&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirichlet+process" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://topicmodels.west.uni-koblenz.de/ckling/tmt/crp.html?parameters=0.5&dp=1#">"Dirichlet Process and Dirichlet Distribution – Polya Restaurant Scheme and Chinese Restaurant Process"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Dirichlet+Process+and+Dirichlet+Distribution+%E2%80%93+Polya+Restaurant+Scheme+and+Chinese+Restaurant+Process&rft_id=http%3A%2F%2Ftopicmodels.west.uni-koblenz.de%2Fckling%2Ftmt%2Fcrp.html%3Fparameters%3D0.5%26dp%3D1%23&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirichlet+process" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">For the proof, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaisley2010" class="citation web cs1">Paisley, John (August 2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150122033202/http://www.columbia.edu/~jwp2128/Teaching/E6892/papers/SimpleProof.pdf">"A simple proof of the stick-breaking construction of the Dirichlet Process"</a> <span class="cs1-format">(PDF)</span>. <i>Columbia University</i>. Archived from <a rel="nofollow" class="external text" href="http://www.columbia.edu/~jwp2128/Teaching/E6892/papers/SimpleProof.pdf">the original</a> <span class="cs1-format">(PDF)</span> on January 22, 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Columbia+University&rft.atitle=A+simple+proof+of+the+stick-breaking+construction+of+the+Dirichlet+Process&rft.date=2010-08&rft.aulast=Paisley&rft.aufirst=John&rft_id=http%3A%2F%2Fwww.columbia.edu%2F~jwp2128%2FTeaching%2FE6892%2Fpapers%2FSimpleProof.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirichlet+process" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAad_van_der_Vaart,_Subhashis_Ghosal2017" class="citation book cs1"><a href="/wiki/Aad_van_der_Vaart" title="Aad van der Vaart">Aad van der Vaart</a>, Subhashis Ghosal (2017). <i>Fundamentals of Bayesian Nonparametric Inference</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-87826-5" title="Special:BookSources/978-0-521-87826-5"><bdi>978-0-521-87826-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Bayesian+Nonparametric+Inference&rft.pub=Cambridge+University+Press&rft.date=2017&rft.isbn=978-0-521-87826-5&rft.au=Aad+van+der+Vaart%2C+Subhashis+Ghosal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirichlet+process" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSudderth2006" class="citation thesis cs1">Sudderth, Erik (2006). <a rel="nofollow" class="external text" href="http://cs.brown.edu/~sudderth/papers/sudderthPhD.pdf"><i>Graphical Models for Visual Object Recognition and Tracking</i></a> <span class="cs1-format">(PDF)</span> (Ph.D.). MIT Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Graphical+Models+for+Visual+Object+Recognition+and+Tracking&rft.inst=MIT+Press&rft.date=2006&rft.aulast=Sudderth&rft.aufirst=Erik&rft_id=http%3A%2F%2Fcs.brown.edu%2F~sudderth%2Fpapers%2FsudderthPhD.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirichlet+process" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNils_Lid_HjortChris_Holmes,_Peter_MüllerStephen_G._Walker2010" class="citation book cs1"><a href="/wiki/Nils_Lid_Hjort" title="Nils Lid Hjort">Nils Lid Hjort</a>; Chris Holmes, Peter Müller; Stephen G. Walker (2010). <i>Bayesian Nonparametrics</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-51346-3" title="Special:BookSources/978-0-521-51346-3"><bdi>978-0-521-51346-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bayesian+Nonparametrics&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-51346-3&rft.au=Nils+Lid+Hjort&rft.au=Chris+Holmes%2C+Peter+M%C3%BCller&rft.au=Stephen+G.+Walker&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirichlet+process" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Sotirios P. Chatzis, "A Latent Variable Gaussian Process Model with Pitman-Yor Process Priors for Multiclass Classification," Neurocomputing, vol. 120, pp. 482–489, Nov. 2013. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.neucom.2013.04.029">10.1016/j.neucom.2013.04.029</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Sotirios P. Chatzis, Yiannis Demiris, "Nonparametric mixtures of Gaussian processes with power-law behaviour," IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 12, pp. 1862–1871, Dec. 2012. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTNNLS.2012.2217986">10.1109/TNNLS.2012.2217986</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRasmussen2000" class="citation journal cs1">Rasmussen, Carl (2000). <a rel="nofollow" class="external text" href="http://www.gatsby.ucl.ac.uk/~edward/pub/inf.mix.nips.99.pdf">"The Infinite Gaussian Mixture Model"</a> <span class="cs1-format">(PDF)</span>. <i>Advances in Neural Information Processing Systems</i>. <b>12</b>: <span class="nowrap">554–</span>560.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Neural+Information+Processing+Systems&rft.atitle=The+Infinite+Gaussian+Mixture+Model&rft.volume=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E554-%3C%2Fspan%3E560&rft.date=2000&rft.aulast=Rasmussen&rft.aufirst=Carl&rft_id=http%3A%2F%2Fwww.gatsby.ucl.ac.uk%2F~edward%2Fpub%2Finf.mix.nips.99.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADirichlet+process" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Sotirios P. Chatzis, Dimitrios Korkinof, and Yiannis Demiris, "A nonparametric Bayesian approach toward robot learning by demonstration," Robotics and Autonomous Systems, vol. 60, no. 6, pp. 789–802, June 2012. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.robot.2012.02.005">10.1016/j.robot.2012.02.005</a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dirichlet_process&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150219021331/https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2010-0006.pdf">Introduction to the Dirichlet Distribution and Related Processes by Frigyik, Kapila and Gupta</a></li> <li><a rel="nofollow" class="external text" href="http://www.gatsby.ucl.ac.uk/~ywteh/research/npbayes/Teh2010a.pdf">Yee Whye Teh's overview of Dirichlet processes</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070524045420/http://www.cs.toronto.edu/~beal/npbayes/">Webpage for the NIPS 2003 workshop on non-parametric Bayesian methods</a></li> <li><a rel="nofollow" class="external text" href="http://www.cs.berkeley.edu/~jordan/nips-tutorial05.ps">Michael Jordan's NIPS 2005 tutorial: <i>Nonparametric Bayesian Methods: Dirichlet Processes, Chinese Restaurant Processes and All That</i></a></li> <li><a rel="nofollow" class="external text" href="http://www.maths.bris.ac.uk/~maxvd/cribsheet.pdf">Peter Green's summary of construction of Dirichlet Processes</a></li> <li><a rel="nofollow" class="external text" href="http://www.stats.bris.ac.uk/~peter/papers/GreenCDP.pdf">Peter Green's paper on probabilistic models of Dirichlet Processes with implications for statistical modelling and analysis</a></li> <li><a rel="nofollow" class="external text" href="http://learning.eng.cam.ac.uk/zoubin/talks/uai05tutorial-b.pdf">Zoubin Ghahramani's UAI 2005 tutorial on Nonparametric Bayesian methods</a></li> <li><a rel="nofollow" class="external text" href="http://ClusterAnalysis.org">GIMM software for performing cluster analysis using Infinite Mixture Models</a></li> <li><a rel="nofollow" class="external text" href="https://archive.today/20121215093339/http://www.ece.sunysb.edu/~zyweng/dpcluster.html">A Toy Example of Clustering using Dirichlet Process.</a> by Zhiyuan Weng</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist 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abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Stochastic_processes" title="Template:Stochastic processes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Stochastic_processes" title="Template talk:Stochastic processes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Stochastic_processes" title="Special:EditPage/Template:Stochastic processes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li> <li><a href="/wiki/Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li> <li><a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Moran_process" title="Moran process">Moran process</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a> <ul><li><a href="/wiki/Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li> <li><a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li> <li><a href="/wiki/Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li> <li><a href="/wiki/Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Additive_process" title="Additive process">Additive process</a></li> <li><a href="/wiki/Bessel_process" title="Bessel process">Bessel process</a></li> <li><a href="/wiki/Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a> <ul><li><a href="/wiki/Birth_process" title="Birth process">pure birth</a></li></ul></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a> <ul><li><a href="/wiki/Brownian_bridge" title="Brownian bridge">Bridge</a></li> <li><a href="/wiki/Brownian_excursion" title="Brownian excursion">Excursion</a></li> <li><a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li> <li><a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li> <li><a href="/wiki/Brownian_meander" title="Brownian meander">Meander</a></li></ul></li> <li><a href="/wiki/Cauchy_process" title="Cauchy process">Cauchy process</a></li> <li><a href="/wiki/Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li> <li><a href="/wiki/Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li> <li><a href="/wiki/Cox_process" title="Cox process">Cox process</a></li> <li><a href="/wiki/Diffusion_process" title="Diffusion process">Diffusion process</a></li> <li><a href="/wiki/Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson Brownian motion</a></li> <li><a href="/wiki/Empirical_process" title="Empirical process">Empirical process</a></li> <li><a href="/wiki/Feller_process" title="Feller process">Feller process</a></li> <li><a href="/wiki/Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li> <li><a href="/wiki/Gamma_process" title="Gamma process">Gamma process</a></li> <li><a href="/wiki/Geometric_process" title="Geometric process">Geometric process</a></li> <li><a href="/wiki/Hawkes_process" title="Hawkes process">Hawkes process</a></li> <li><a href="/wiki/Hunt_process" title="Hunt process">Hunt process</a></li> <li><a href="/wiki/Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li> <li><a href="/wiki/It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li> <li><a href="/wiki/It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li> <li><a href="/wiki/Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li> <li><a href="/wiki/Jump_process" title="Jump process">Jump process</a></li> <li><a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy process</a></li> <li><a href="/wiki/Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li> <li><a href="/wiki/Markov_additive_process" title="Markov additive process">Markov additive process</a></li> <li><a href="/wiki/McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li> <li><a href="/wiki/Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson process</a> <ul><li><a href="/wiki/Compound_Poisson_process" title="Compound Poisson process">Compound</a></li> <li><a href="/wiki/Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li> <li><a href="/wiki/Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li> <li><a href="/wiki/Semimartingale" title="Semimartingale">Semimartingale</a></li> <li><a href="/wiki/Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li> <li><a href="/wiki/Stable_process" title="Stable process">Stable process</a></li> <li><a href="/wiki/Superprocess" title="Superprocess">Superprocess</a></li> <li><a href="/wiki/Telegraph_process" title="Telegraph process">Telegraph process</a></li> <li><a href="/wiki/Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a></li> <li><a href="/wiki/Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian process</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li> <li><a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li> <li><a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a> <ul><li><a href="/wiki/Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li> <li><a href="/wiki/Local_martingale" title="Local martingale">Local</a></li> <li><a href="/wiki/Submartingale" class="mw-redirect" title="Submartingale">Sub-</a></li> <li><a href="/wiki/Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li> <li><a href="/wiki/Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li> <li><a href="/wiki/Regenerative_process" title="Regenerative process">Regenerative process</a></li> <li><a href="/wiki/Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li> <li><a href="/wiki/Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li> <li><a href="/wiki/White_noise" title="White noise">White noise</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Dirichlet process</a></li> <li><a href="/wiki/Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li> <li><a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li> <li><a href="/wiki/Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li> <li><a href="/wiki/Ising_model" title="Ising model">Ising model</a> <ul><li><a href="/wiki/Potts_model" title="Potts model">Potts model</a></li> <li><a href="/wiki/Boolean_network" title="Boolean network">Boolean network</a></li></ul></li> <li><a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation</a></li> <li><a href="/wiki/Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li> <li><a href="/wiki/Point_process" title="Point process">Point process</a> <ul><li><a href="/wiki/Point_process#Cox_point_process" title="Point process">Cox</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li> <li><a href="/wiki/Random_field" title="Random field">Random field</a></li> <li><a href="/wiki/Random_graph" title="Random graph">Random graph</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li> <li><a href="/wiki/Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li> <li><a href="/wiki/Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li> <li><a href="/wiki/Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li> <li><a href="/wiki/Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li> <li><a href="/wiki/Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li> <li><a href="/wiki/Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li> <li><a href="/wiki/Chen_model" title="Chen model">Chen</a></li> <li><a href="/wiki/Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li> <li><a href="/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li> <li><a href="/wiki/Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li> <li><a href="/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li> <li><a href="/wiki/Heston_model" title="Heston model">Heston</a></li> <li><a href="/wiki/Ho%E2%80%93Lee_model" title="Ho–Lee model">Ho–Lee</a></li> <li><a href="/wiki/Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li> <li><a href="/wiki/Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li> <li><a href="/wiki/LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li> <li><a href="/wiki/Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li> <li><a href="/wiki/SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li> <li><a href="/wiki/Vasicek_model" title="Vasicek model">Vašíček</a></li> <li><a href="/wiki/Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li> <li><a href="/wiki/Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li> <li><a href="/wiki/Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li> <li><a href="/wiki/Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bulk_queue" title="Bulk queue">Bulk</a></li> <li><a href="/wiki/Fluid_queue" title="Fluid queue">Fluid</a></li> <li><a href="/wiki/G-network" title="G-network">Generalized queueing network</a></li> <li><a href="/wiki/M/G/1_queue" title="M/G/1 queue">M/G/1</a></li> <li><a href="/wiki/M/M/1_queue" title="M/M/1 queue">M/M/1</a></li> <li><a href="/wiki/M/M/c_queue" title="M/M/c queue">M/M/c</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li> <li><a href="/wiki/Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li> <li><a href="/wiki/Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li> <li><a href="/wiki/Ergodicity" title="Ergodicity">Ergodic</a></li> <li><a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li> <li><a href="/wiki/Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li> <li><a href="/wiki/Markov_property" title="Markov property">Markov</a></li> <li><a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li> <li><a href="/wiki/Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li> <li><a href="/wiki/Predictable_process" title="Predictable process">Predictable</a></li> <li><a href="/wiki/Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li> <li><a href="/wiki/Self-similar_process" title="Self-similar process">Self-similar</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationary</a></li> <li><a href="/wiki/Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Donsker%27s_theorem" title="Donsker's theorem">Donsker's theorem</a></li> <li><a href="/wiki/Doob%27s_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li> <li><a href="/wiki/Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li> <li><a href="/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li> <li><a href="/wiki/Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li> <li><a href="/wiki/Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li> <li><a href="/wiki/Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li> <li><a href="/wiki/Sanov%27s_theorem" title="Sanov's theorem">Sanov's theorem</a></li> <li><a href="/wiki/Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="/wiki/Blumenthal%27s_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="/wiki/Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="/wiki/Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="/wiki/Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="/wiki/L%C3%A9vy%27s_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li> <li><a href="/wiki/Doob%27s_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li> <li><a href="/wiki/Doob%27s_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li> <li><a href="/wiki/Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li> <li><a href="/wiki/Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li> <li><a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li> <li><a href="/wiki/Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li> <li><a href="/wiki/Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li> <li><a href="/wiki/Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li> <li><a href="/wiki/Doob%27s_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li> <li><a href="/wiki/Dynkin%27s_formula" title="Dynkin's formula">Dynkin's formula</a></li> <li><a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li> <li><a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li> <li><a href="/wiki/Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li> <li><a href="/wiki/Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li> <li><a href="/wiki/It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li> <li><a href="/wiki/It%C3%B4%27s_lemma" title="Itô's lemma">Itô's lemma</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li> <li><a href="/wiki/Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li> <li><a href="/wiki/Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li> <li><a href="/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li> <li><a href="/wiki/Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li> <li><a href="/wiki/Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li> <li><a href="/wiki/Prokhorov%27s_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li> <li><a href="/wiki/Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li> <li><a href="/wiki/Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li> <li><a href="/wiki/Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li> <li><a href="/wiki/Skorokhod%27s_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li> <li><a href="/wiki/Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li> <li><a href="/wiki/Snell_envelope" title="Snell envelope">Snell envelope</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a> <ul><li><a href="/wiki/Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li> <li><a href="/wiki/Stopping_time" title="Stopping time">Stopping time</a></li> <li><a href="/wiki/Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li> <li><a href="/wiki/Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li> <li><a href="/wiki/Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li> <li><a href="/wiki/Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical</a></li> <li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li> <li><a href="/wiki/Stochastic_control" title="Stochastic control">Control theory</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li> <li><a href="/wiki/Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li> <li><a href="/wiki/Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a></li> <li><a href="/wiki/Renewal_theory" title="Renewal theory">Renewal theory</a></li> <li><a href="/wiki/Ruin_theory" title="Ruin theory">Ruin theory</a></li> <li><a href="/wiki/Signal_processing" title="Signal processing">Signal processing</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li> <li><a href="/wiki/Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li> <li><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li> <li><a href="/wiki/Category:Stochastic_processes" title="Category:Stochastic processes">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" 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Dirichlet process - Wikipedia