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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>AlgebraicInference.jl · AlgebraicInference.jl</title><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>AlgebraicInference.jl</a></span></div><form class="docs-search" action="search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>AlgebraicInference.jl</a><ul class="internal"><li><a class="tocitem" href="#Gaussian-Systems"><span>Gaussian Systems</span></a></li><li><a class="tocitem" href="#Hypergraph-Categories"><span>Hypergraph Categories</span></a></li><li><a class="tocitem" href="#Message-Passing"><span>Message Passing</span></a></li></ul></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="generated/regression/">Linear Regression</a></li><li><a class="tocitem" href="generated/kalman/">Kalman Filter</a></li></ul></li><li><a class="tocitem" href="api/">Library Reference</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>AlgebraicInference.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>AlgebraicInference.jl</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/samuelsonric/AlgebraicInference.jl/blob/master/docs/src/index.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="AlgebraicInference.jl"><a class="docs-heading-anchor" href="#AlgebraicInference.jl">AlgebraicInference.jl</a><a id="AlgebraicInference.jl-1"></a><a class="docs-heading-anchor-permalink" href="#AlgebraicInference.jl" title="Permalink"></a></h1><p>AlgebraicInference.jl is a library for performing Bayesian inference on wiring diagrams, building on <a href="https://algebraicjulia.github.io/Catlab.jl/dev/">Catlab.jl</a>.</p><h2 id="Gaussian-Systems"><a class="docs-heading-anchor" href="#Gaussian-Systems">Gaussian Systems</a><a id="Gaussian-Systems-1"></a><a class="docs-heading-anchor-permalink" href="#Gaussian-Systems" title="Permalink"></a></h2><p>Gaussian systems were introduced by Jan Willems in his 2013 article <em>Open Stochastic Systems</em>. A probability space <span>$\Sigma = (\mathbb{R}^n, \mathcal{E}, P)$</span> is called an <span>$n$</span>-variate Gaussian system with fiber <span>$\mathbb{L} \subseteq \mathbb{R}^n$</span> if it is isomorphic to a Gaussian measure on the quotient space <span>$\mathbb{R}^n / \mathbb{L}$</span>.</p><p>If <span>$\mathbb{L} = \{0\}$</span>, then <span>$\Sigma$</span> is an <span>$n$</span>-variate normal distribution.</p><p>Every <span>$n$</span>-variate Gaussian system <span>$\Sigma$</span> corresponds to a convex <em>energy function</em> <span>$E: \mathbb{R}^n \to (-\infty, \infty]$</span> of the form</p><p class="math-container">\[ E(x) = \begin{cases}
\frac{1}{2} x^\mathsf{T} P x - x^\mathsf{T} p &amp; Sx = s \\
\infty &amp; \text{else},
\end{cases}\]</p><p>where <span>$P$</span> and <span>$S$</span> are positive semidefinite matrices, <span>$p \in \mathtt{image}(P)$</span>, and <span>$s \in \mathtt{image}(S)$</span>.</p><p>If <span>$\Sigma$</span> is an <span>$n$</span>-variate normal distribution, then <span>$E$</span> is its negative log-density.</p><h2 id="Hypergraph-Categories"><a class="docs-heading-anchor" href="#Hypergraph-Categories">Hypergraph Categories</a><a id="Hypergraph-Categories-1"></a><a class="docs-heading-anchor-permalink" href="#Hypergraph-Categories" title="Permalink"></a></h2><p>There exists a hypergraph PROP whose morphisms <span>$m \to n$</span> are <span>$m + n$</span>-variate Gaussian systems. Hence, Gaussian systems can be composed using undirected wiring diagrams.</p><p><img src="inference.svg" alt="inference"/></p><p>These wiring diagrams look a lot like <a href="https://en.wikipedia.org/wiki/Graphical_model">undirected graphical models</a>. One difference is that wiring diagrams can contain half-edges, which specify which variables are marginalized out during composition. Hence, a wiring diagram can be thought of as an <em>inference problem</em>: a graphical model paired with a query.</p><h2 id="Message-Passing"><a class="docs-heading-anchor" href="#Message-Passing">Message Passing</a><a id="Message-Passing-1"></a><a class="docs-heading-anchor-permalink" href="#Message-Passing" title="Permalink"></a></h2><p>Bayesian inference problems on large graphs are often solved using <a href="https://en.wikipedia.org/wiki/Belief_propagation">message passing</a>. With AlgebraicInference.jl you can compose large numbers of Gaussian systems by translating undirected wiring diagrams into inference problems over a <a href="https://en.wikipedia.org/wiki/Information_algebra">valuation algebra</a>. These problems can be solved using generic inference algorithms like the Shenoy-Shafer architecture.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="generated/regression/">Linear Regression »</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="Thursday 21 September 2023 02:03">Thursday 21 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
\end{cases}\]</p><p>where <span>$P$</span> and <span>$S$</span> are positive semidefinite matrices, <span>$p \in \mathtt{image}(P)$</span>, and <span>$s \in \mathtt{image}(S)$</span>.</p><p>If <span>$\Sigma$</span> is an <span>$n$</span>-variate normal distribution, then <span>$E$</span> is its negative log-density.</p><h2 id="Hypergraph-Categories"><a class="docs-heading-anchor" href="#Hypergraph-Categories">Hypergraph Categories</a><a id="Hypergraph-Categories-1"></a><a class="docs-heading-anchor-permalink" href="#Hypergraph-Categories" title="Permalink"></a></h2><p>There exists a hypergraph PROP whose morphisms <span>$m \to n$</span> are <span>$m + n$</span>-variate Gaussian systems. Hence, Gaussian systems can be composed using undirected wiring diagrams.</p><p><img src="inference.svg" alt="inference"/></p><p>These wiring diagrams look a lot like <a href="https://en.wikipedia.org/wiki/Graphical_model">undirected graphical models</a>. One difference is that wiring diagrams can contain half-edges, which specify which variables are marginalized out during composition. Hence, a wiring diagram can be thought of as an <em>inference problem</em>: a graphical model paired with a query.</p><h2 id="Message-Passing"><a class="docs-heading-anchor" href="#Message-Passing">Message Passing</a><a id="Message-Passing-1"></a><a class="docs-heading-anchor-permalink" href="#Message-Passing" title="Permalink"></a></h2><p>Bayesian inference problems on large graphs are often solved using <a href="https://en.wikipedia.org/wiki/Belief_propagation">message passing</a>. With AlgebraicInference.jl you can compose large numbers of Gaussian systems by translating undirected wiring diagrams into inference problems over a <a href="https://en.wikipedia.org/wiki/Information_algebra">valuation algebra</a>. These problems can be solved using generic inference algorithms like the Shenoy-Shafer architecture.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="generated/regression/">Linear Regression »</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="Thursday 21 September 2023 02:11">Thursday 21 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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