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</li><li>
<div id="Rabiner1989">Rabiner, L. (1989). <i>A tutorial on hidden Markov models and selected applications in speech recognition</i>. <a href='https://doi.org/10.1109/5.18626'>Proceedings of the IEEE <b>77</b>, 257–286</a>.</div>
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</ul></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../api/">API reference »</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="Tuesday 12 September 2023 18:47">Tuesday 12 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
</ul></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../api/">API reference »</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="Wednesday 13 September 2023 17:07">Wednesday 13 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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<!DOCTYPE html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Debugging · HiddenMarkovModels.jl</title><script data-outdated-warner src="../assets/warner.js"></script><link rel="canonical" href="https://gdalle.github.io/HiddenMarkovModels.jl/debugging/"/><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"><a class="docs-logo" href="../"><img src="../assets/logo.png" alt="HiddenMarkovModels.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">HiddenMarkovModels.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><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Essentials</span><ul><li><a class="tocitem" href="../background/">Background</a></li><li><a class="tocitem" href="../api/">API reference</a></li></ul></li><li><span class="tocitem">Tutorials</span><ul><li><a class="tocitem" href="../tuto_builtin/">Built-in HMM</a></li><li><a class="tocitem" href="../tuto_custom/">Custom HMM</a></li><li class="is-active"><a class="tocitem" href>Debugging</a><ul class="internal"><li><a class="tocitem" href="#Numerical-overflow"><span>Numerical overflow</span></a></li></ul></li></ul></li><li><span class="tocitem">Alternatives</span><ul><li><a class="tocitem" href="../alt_features/">Features</a></li><li><a class="tocitem" href="../alt_performance/">Performance</a></li></ul></li><li><span class="tocitem">Advanced</span><ul><li><a class="tocitem" href="../formulas/">Formulas</a></li><li><a class="tocitem" href="../roadmap/">Roadmap</a></li></ul></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><a class="is-disabled">Tutorials</a></li><li class="is-active"><a href>Debugging</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Debugging</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/gdalle/HiddenMarkovModels.jl/blob/main/docs/src/debugging.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="Debugging"><a class="docs-heading-anchor" href="#Debugging">Debugging</a><a id="Debugging-1"></a><a class="docs-heading-anchor-permalink" href="#Debugging" title="Permalink"></a></h1><h2 id="Numerical-overflow"><a class="docs-heading-anchor" href="#Numerical-overflow">Numerical overflow</a><a id="Numerical-overflow-1"></a><a class="docs-heading-anchor-permalink" href="#Numerical-overflow" title="Permalink"></a></h2><p>The most frequent error you will encounter is an <code>OverflowError</code> during forward-backward, telling you that &quot;some values are infinite / NaN&quot;. This can happen for a variety of reasons, so here are a few leads worth investigating:</p><ul><li>Increase the duration of the sequence / the number of sequences (to get more data)</li><li>Reduce the number of states (to make every one of them useful)</li><li>Add a prior to your transition matrix / observation distributions (to avoid degenerate behavior like zero variance in a Gaussian)</li><li>Pick a better initialization (to start closer to the supposed ground truth)</li><li>Use <a href="https://github.com/cjdoris/LogarithmicNumbers.jl">LogarithmicNumbers.jl</a> in strategic places (to guarantee numerical stability). Note that these numbers don&#39;t play nicely with Distributions.jl, so you may have to roll out your own observation distribution.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../tuto_custom/">« Custom HMM</a><a class="docs-footer-nextpage" href="../alt_features/">Features »</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="Tuesday 12 September 2023 18:47">Tuesday 12 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Debugging · HiddenMarkovModels.jl</title><script data-outdated-warner src="../assets/warner.js"></script><link rel="canonical" href="https://gdalle.github.io/HiddenMarkovModels.jl/debugging/"/><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"><a class="docs-logo" href="../"><img src="../assets/logo.png" alt="HiddenMarkovModels.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">HiddenMarkovModels.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><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Essentials</span><ul><li><a class="tocitem" href="../background/">Background</a></li><li><a class="tocitem" href="../api/">API reference</a></li></ul></li><li><span class="tocitem">Tutorials</span><ul><li><a class="tocitem" href="../tuto_builtin/">Built-in HMM</a></li><li><a class="tocitem" href="../tuto_custom/">Custom HMM</a></li><li class="is-active"><a class="tocitem" href>Debugging</a><ul class="internal"><li><a class="tocitem" href="#Numerical-overflow"><span>Numerical overflow</span></a></li></ul></li></ul></li><li><span class="tocitem">Alternatives</span><ul><li><a class="tocitem" href="../alt_features/">Features</a></li><li><a class="tocitem" href="../alt_performance/">Performance</a></li></ul></li><li><span class="tocitem">Advanced</span><ul><li><a class="tocitem" href="../formulas/">Formulas</a></li><li><a class="tocitem" href="../roadmap/">Roadmap</a></li></ul></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><a class="is-disabled">Tutorials</a></li><li class="is-active"><a href>Debugging</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Debugging</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/gdalle/HiddenMarkovModels.jl/blob/main/docs/src/debugging.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="Debugging"><a class="docs-heading-anchor" href="#Debugging">Debugging</a><a id="Debugging-1"></a><a class="docs-heading-anchor-permalink" href="#Debugging" title="Permalink"></a></h1><h2 id="Numerical-overflow"><a class="docs-heading-anchor" href="#Numerical-overflow">Numerical overflow</a><a id="Numerical-overflow-1"></a><a class="docs-heading-anchor-permalink" href="#Numerical-overflow" title="Permalink"></a></h2><p>The most frequent error you will encounter is an <code>OverflowError</code> during forward-backward, telling you that &quot;some values are infinite / NaN&quot;. This can happen for a variety of reasons, so here are a few leads worth investigating:</p><ul><li>Increase the duration of the sequence / the number of sequences (to get more data)</li><li>Reduce the number of states (to make every one of them useful)</li><li>Add a prior to your transition matrix / observation distributions (to avoid degenerate behavior like zero variance in a Gaussian)</li><li>Pick a better initialization (to start closer to the supposed ground truth)</li><li>Use <a href="https://github.com/cjdoris/LogarithmicNumbers.jl">LogarithmicNumbers.jl</a> in strategic places (to guarantee numerical stability). Note that these numbers don&#39;t play nicely with Distributions.jl, so you may have to roll out your own observation distribution.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../tuto_custom/">« Custom HMM</a><a class="docs-footer-nextpage" href="../alt_features/">Features »</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="Wednesday 13 September 2023 17:07">Wednesday 13 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
2 changes: 1 addition & 1 deletion dev/formulas/index.html
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\frac{\partial \log \mathcal{L}}{\partial a_{i,j}} &amp;= \sum_{t=1}^{T-1} \bar{\alpha}_{i,t} \frac{b_{j,t+1}}{m_{t+1}} \bar{\beta}_{j,t+1} \\
\frac{\partial \log \mathcal{L}}{\partial \log b_{j,1}} &amp;= \pi_j \frac{b_{j,1}}{m_1} \bar{\beta}_{j,1} = \frac{\bar{\alpha}_{j,1} \bar{\beta}_{j,1}}{c_1} = \gamma_{j,1} \\
\frac{\partial \log \mathcal{L}}{\partial \log b_{j,t}} &amp;= \sum_{i=1}^N \bar{\alpha}_{i,t-1} a_{i,j} \frac{b_{j,t}}{m_t} \bar{\beta}_{j,t} = \frac{\bar{\alpha}_{j,t} \bar{\beta}_{j,t}}{c_t} = \gamma_{j,t}
\end{align*}\]</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../alt_performance/">« Performance</a><a class="docs-footer-nextpage" href="../roadmap/">Roadmap »</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="Tuesday 12 September 2023 18:47">Tuesday 12 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
\end{align*}\]</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../alt_performance/">« Performance</a><a class="docs-footer-nextpage" href="../roadmap/">Roadmap »</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="Wednesday 13 September 2023 17:07">Wednesday 13 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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