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 \setminus V_i} \prod_{\phi \in \bm{\phi}} \phi(V^{\prime\prime},\bm{e}) }{
     PR(\bm{V}^{\prime} \mid \bm{E}=\bm{e}) }\]</p><p>This task involves computing the marginal probability of a subset of variables, integrating out the others. In other words, it computes the probability distribution of some variables of interest regardless of the states of all other variables. This is useful when we&#39;re interested in the probabilities of some specific variables in the model, but not the entire model.</p><h3 id="Maximum-a-Posteriori-Probability-estimation-(MAP)"><a class="docs-heading-anchor" href="#Maximum-a-Posteriori-Probability-estimation-(MAP)">Maximum a Posteriori Probability estimation (MAP)</a><a id="Maximum-a-Posteriori-Probability-estimation-(MAP)-1"></a><a class="docs-heading-anchor-permalink" href="#Maximum-a-Posteriori-Probability-estimation-(MAP)" title="Permalink"></a></h3><p>Computing the most likely assignment to all variables given evidence:</p><p class="math-container">\[MAP(V_i \mid \bm{E}=\bm{e}) = \arg \max_{V^{\prime} \in \bm{V}^{\prime}}
 \prod_{\phi \in \bm{\phi}} \phi(V^{\prime},\bm{e})\]</p><p>In the MAP task, given some observed variables, the goal is to find the most probable assignment of values to some subset of the unobserved variables. It provides the states of variables that maximize the posterior probability given some observed evidence. This is often used when we want the most likely explanation or prediction according to the model.</p><h3 id="Marginal-Maximum-a-Posteriori-(MMAP)"><a class="docs-heading-anchor" href="#Marginal-Maximum-a-Posteriori-(MMAP)">Marginal Maximum a Posteriori (MMAP)</a><a id="Marginal-Maximum-a-Posteriori-(MMAP)-1"></a><a class="docs-heading-anchor-permalink" href="#Marginal-Maximum-a-Posteriori-(MMAP)" title="Permalink"></a></h3><p>Computing the most likely assignment to the query variables, <span>$\bm{Q} \subset \bm{V}^{\prime}$</span> after marginalizing out the remaining variables <span>$\bm{Z} = \bm{V}^{\prime} \setminus \bm{Q}$</span>, also known as <em>hidden</em> or <em>latent</em> variables:</p><p class="math-container">\[MMAP(V_i \mid \bm{E}=e) = \arg \max_{Q \in \bm{Q}} \sum_{Z \in \bm{Z}}
-\prod_{\phi \in \bm{\phi}} \phi(Q, Z, e)\]</p><p>This task is essentially a combination of the MAR and MAP tasks. The MMAP task involves finding the most probable assignment (the MAP estimate) for a subset of the variables, while marginalizing over (summing out) the remaining variables. This task is useful when we want to know the most likely state of some variables, but there&#39;s some uncertainty over others that we need to average out.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-lauritzen1988local"><a class="tag is-link" href="#citeref-lauritzen1988local">lauritzen1988local</a>Steffen L Lauritzen and David J Spiegelhalter. Local computations with probabilities on graphical structures and their application to expert systems. <em>Journal of the Royal Statistical Society: Series B (Methodological)</em>, 50(2):157–194, 1988.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../examples-overview/">Overview »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 24 July 2023 03:30">Monday 24 July 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\prod_{\phi \in \bm{\phi}} \phi(Q, Z, e)\]</p><p>This task is essentially a combination of the MAR and MAP tasks. The MMAP task involves finding the most probable assignment (the MAP estimate) for a subset of the variables, while marginalizing over (summing out) the remaining variables. This task is useful when we want to know the most likely state of some variables, but there&#39;s some uncertainty over others that we need to average out.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-lauritzen1988local"><a class="tag is-link" href="#citeref-lauritzen1988local">lauritzen1988local</a>Steffen L Lauritzen and David J Spiegelhalter. Local computations with probabilities on graphical structures and their application to expert systems. <em>Journal of the Royal Statistical Society: Series B (Methodological)</em>, 50(2):157–194, 1988.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../examples-overview/">Overview »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 14 August 2023 08:36">Monday 14 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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