radu
/
SSMBook
spegling av https://github.com/probml/ssm-book.git


			
							123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551552553554555556557558559560561562563564565566567568569570571572573574575576577578579580581582583584585586587588589590591592593594595596
							
<!DOCTYPE html>

<html>
  <head>
    <meta charset="utf-8" />
    <meta name="viewport" content="width=device-width, initial-scale=1.0" />
    <title>Parameter estimation (learning) &#8212; State Space Models: A Modern Approach</title>
    
  <link href="../../_static/css/theme.css" rel="stylesheet">
  <link href="../../_static/css/index.ff1ffe594081f20da1ef19478df9384b.css" rel="stylesheet">

    
  <link rel="stylesheet"
    href="../../_static/vendor/fontawesome/5.13.0/css/all.min.css">
  <link rel="preload" as="font" type="font/woff2" crossorigin
    href="../../_static/vendor/fontawesome/5.13.0/webfonts/fa-solid-900.woff2">
  <link rel="preload" as="font" type="font/woff2" crossorigin
    href="../../_static/vendor/fontawesome/5.13.0/webfonts/fa-brands-400.woff2">

    
    <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
    <link rel="stylesheet" type="text/css" href="../../_static/sphinx-book-theme.css?digest=c3fdc42140077d1ad13ad2f1588a4309" />
    <link rel="stylesheet" type="text/css" href="../../_static/togglebutton.css" />
    <link rel="stylesheet" type="text/css" href="../../_static/copybutton.css" />
    <link rel="stylesheet" type="text/css" href="../../_static/mystnb.css" />
    <link rel="stylesheet" type="text/css" href="../../_static/sphinx-thebe.css" />
    <link rel="stylesheet" type="text/css" href="../../_static/panels-main.c949a650a448cc0ae9fd3441c0e17fb0.css" />
    <link rel="stylesheet" type="text/css" href="../../_static/panels-variables.06eb56fa6e07937060861dad626602ad.css" />
    
  <link rel="preload" as="script" href="../../_static/js/index.be7d3bbb2ef33a8344ce.js">

    <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
    <script src="../../_static/jquery.js"></script>
    <script src="../../_static/underscore.js"></script>
    <script src="../../_static/doctools.js"></script>
    <script src="../../_static/clipboard.min.js"></script>
    <script src="../../_static/copybutton.js"></script>
    <script>let toggleHintShow = 'Click to show';</script>
    <script>let toggleHintHide = 'Click to hide';</script>
    <script>let toggleOpenOnPrint = 'true';</script>
    <script src="../../_static/togglebutton.js"></script>
    <script>var togglebuttonSelector = '.toggle, .admonition.dropdown, .tag_hide_input div.cell_input, .tag_hide-input div.cell_input, .tag_hide_output div.cell_output, .tag_hide-output div.cell_output, .tag_hide_cell.cell, .tag_hide-cell.cell';</script>
    <script src="../../_static/sphinx-book-theme.d59cb220de22ca1c485ebbdc042f0030.js"></script>
    <script>const THEBE_JS_URL = "https://unpkg.com/thebe@0.8.2/lib/index.js"
const thebe_selector = ".thebe,.cell"
const thebe_selector_input = "pre"
const thebe_selector_output = ".output, .cell_output"
</script>
    <script async="async" src="../../_static/sphinx-thebe.js"></script>
    <script>window.MathJax = {"tex": {"macros": {"covMat": "\\boldsymbol{\\Sigma}", "data": "\\mathcal{D}", "defeq": "\\triangleq", "diag": "\\mathrm{diag}", "discreteState": "s", "dotstar": "\\odot", "dynamicsFn": "\\mathbf{f}", "floor": ["\\lfloor#1\\rfloor", 1], "gainMatrix": "\\mathbf{K}", "gainMatrixReverse": "\\mathbf{G}", "gauss": "\\mathcal{N}", "gaussInfo": "\\mathcal{N}_{\\text{info}}", "hidden": "\\mathbf{x}", "hiddenScalar": "x", "hmmInit": "\\boldsymbol{\\pi}", "hmmInitScalar": "\\pi", "hmmObs": "\\mathbf{B}", "hmmObsScalar": "B", "hmmTrans": "\\mathbf{A}", "hmmTransScalar": "A", "infoMat": "\\precMat", "input": "\\mathbf{u}", "inputs": "\\input", "inv": ["{#1}^{-1}", 1], "keyword": ["\\textbf{#1}", 1], "ldsDyn": "\\mathbf{F}", "ldsDynIn": "\\mathbf{B}", "initMean": "\\boldsymbol{\\mean}_0", "initCov": "\\boldsymbol{\\covMat}_0", "ldsObs": "\\mathbf{H}", "ldsObsIn": "\\mathbf{D}", "ldsTrans": "\\ldsDyn", "ldsTransIn": "\\ldsDynIn", "obsCov": "\\mathbf{R}", "obsNoise": "\\boldsymbol{r}", "map": "\\mathrm{map}", "measurementFn": "\\mathbf{h}", "mean": "\\boldsymbol{\\mu}", "mle": "\\mathrm{mle}", "nlatents": "n_x", "nhidden": "\\nlatents", "ninputs": "n_u", "nobs": "n_y", "nsymbols": "n_y", "nstates": "n_s", "obs": "\\mathbf{y}", "obsScalar": "y", "observed": "\\obs", "obsFn": "\\measurementFn", "params": "\\boldsymbol{\\theta}", "precMean": "\\boldsymbol{\\eta}", "precMat": "\\boldsymbol{\\Lambda}", "real": "\\mathbb{R}", "sigmoid": "\\sigma", "softmax": "\\boldsymbol{\\sigma}", "trans": "\\mathsf{T}", "transpose": ["{#1}^{\\trans}", 1], "transCov": "\\mathbf{Q}", "transNoise": "\\mathbf{q}", "valpha": "\\boldsymbol{\\alpha}", "vbeta": "\\boldsymbol{\\beta}", "vdelta": "\\boldsymbol{\\delta}", "vepsilon": "\\boldsymbol{\\epsilon}", "vlambda": "\\boldsymbol{\\lambda}", "vLambda": "\\boldsymbol{\\Lambda}", "vmu": "\\boldsymbol{\\mu}", "vpi": "\\boldsymbol{\\pi}", "vsigma": "\\boldsymbol{\\sigma}", "vSigma": "\\boldsymbol{\\Sigma}", "vone": "\\boldsymbol{1}", "vzero": "\\boldsymbol{0}"}}, "options": {"processHtmlClass": "tex2jax_process|mathjax_process|math|output_area"}}</script>
    <script defer="defer" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
    <link rel="index" title="Index" href="../../genindex.html" />
    <link rel="search" title="Search" href="../../search.html" />
    <link rel="next" title="Hidden Markov Models" href="../hmm/hmm_index.html" />
    <link rel="prev" title="States estimation (inference)" href="inference.html" />
    <meta name="viewport" content="width=device-width, initial-scale=1" />
    <meta name="docsearch:language" content="None">
    

    <!-- Google Analytics -->
    
  </head>
  <body data-spy="scroll" data-target="#bd-toc-nav" data-offset="80">
    
    <div class="container-fluid" id="banner"></div>

    
    <div class="container-xl">
      <div class="row">
          
<div class="col-12 col-md-3 bd-sidebar site-navigation show" id="site-navigation">
    
        <div class="navbar-brand-box">
    <a class="navbar-brand text-wrap" href="../../index.html">
      
      
      <h1 class="site-logo" id="site-title">State Space Models: A Modern Approach</h1>
      
    </a>
</div><form class="bd-search d-flex align-items-center" action="../../search.html" method="get">
  <i class="icon fas fa-search"></i>
  <input type="search" class="form-control" name="q" id="search-input" placeholder="Search this book..." aria-label="Search this book..." autocomplete="off" >
</form><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
    <div class="bd-toc-item active">
        <ul class="nav bd-sidenav">
 <li class="toctree-l1">
  <a class="reference internal" href="../../root.html">
   State Space Models: A Modern Approach
  </a>
 </li>
</ul>
<ul class="current nav bd-sidenav">
 <li class="toctree-l1 current active has-children">
  <a class="reference internal" href="ssm_index.html">
   State Space Models
  </a>
  <input checked="" class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/>
  <label for="toctree-checkbox-1">
   <i class="fas fa-chevron-down">
   </i>
  </label>
  <ul class="current">
   <li class="toctree-l2">
    <a class="reference internal" href="ssm_intro.html">
     What are State Space Models?
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="hmm.html">
     Hidden Markov Models
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="lds.html">
     Linear Gaussian SSMs
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="nlds.html">
     Nonlinear Gaussian SSMs
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="inference.html">
     States estimation (inference)
    </a>
   </li>
   <li class="toctree-l2 current active">
    <a class="current reference internal" href="#">
     Parameter estimation (learning)
    </a>
   </li>
  </ul>
 </li>
 <li class="toctree-l1 has-children">
  <a class="reference internal" href="../hmm/hmm_index.html">
   Hidden Markov Models
  </a>
  <input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/>
  <label for="toctree-checkbox-2">
   <i class="fas fa-chevron-down">
   </i>
  </label>
  <ul>
   <li class="toctree-l2">
    <a class="reference internal" href="../hmm/hmm_filter.html">
     HMM filtering (forwards algorithm)
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="../hmm/hmm_smoother.html">
     HMM smoothing (forwards-backwards algorithm)
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="../hmm/hmm_viterbi.html">
     Viterbi algorithm
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="../hmm/hmm_parallel.html">
     Parallel HMM  smoothing
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="../hmm/hmm_sampling.html">
     Forwards-filtering backwards-sampling algorithm
    </a>
   </li>
  </ul>
 </li>
 <li class="toctree-l1 has-children">
  <a class="reference internal" href="../lgssm/lgssm_index.html">
   Linear-Gaussian SSMs
  </a>
  <input class="toctree-checkbox" id="toctree-checkbox-3" name="toctree-checkbox-3" type="checkbox"/>
  <label for="toctree-checkbox-3">
   <i class="fas fa-chevron-down">
   </i>
  </label>
  <ul>
   <li class="toctree-l2">
    <a class="reference internal" href="../lgssm/kalman_filter.html">
     Kalman filtering
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="../lgssm/kalman_smoother.html">
     Kalman (RTS) smoother
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="../lgssm/kalman_parallel.html">
     Parallel Kalman Smoother
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="../lgssm/kalman_sampling.html">
     Forwards-filtering backwards sampling
    </a>
   </li>
  </ul>
 </li>
 <li class="toctree-l1 has-children">
  <a class="reference internal" href="../extended/extended_index.html">
   Extended (linearized) methods
  </a>
  <input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/>
  <label for="toctree-checkbox-4">
   <i class="fas fa-chevron-down">
   </i>
  </label>
  <ul>
   <li class="toctree-l2">
    <a class="reference internal" href="../extended/extended_filter.html">
     Extended Kalman filtering
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="../extended/extended_smoother.html">
     Extended Kalman smoother
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="../extended/extended_parallel.html">
     Parallel extended Kalman smoothing
    </a>
   </li>
  </ul>
 </li>
 <li class="toctree-l1 has-children">
  <a class="reference internal" href="../unscented/unscented_index.html">
   Unscented methods
  </a>
  <input class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/>
  <label for="toctree-checkbox-5">
   <i class="fas fa-chevron-down">
   </i>
  </label>
  <ul>
   <li class="toctree-l2">
    <a class="reference internal" href="../unscented/unscented_filter.html">
     Unscented filtering
    </a>
   </li>
   <li class="toctree-l2">
    <a class="reference internal" href="../unscented/unscented_smoother.html">
     Unscented smoothing
    </a>
   </li>
  </ul>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../quadrature/quadrature_index.html">
   Quadrature and cubature methods
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../postlin/postlin_index.html">
   Posterior linearization
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../adf/adf_index.html">
   Assumed Density Filtering
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../vi/vi_index.html">
   Variational inference
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../pf/pf_index.html">
   Particle filtering
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../smc/smc_index.html">
   Sequential Monte Carlo
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../learning/learning_index.html">
   Offline parameter estimation (learning)
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../tracking/tracking_index.html">
   Multi-target tracking
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../ensemble/ensemble_index.html">
   Data assimilation using Ensemble Kalman filter
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../bnp/bnp_index.html">
   Bayesian non-parametric SSMs
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../changepoint/changepoint_index.html">
   Changepoint detection
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../timeseries/timeseries_index.html">
   Timeseries forecasting
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../gp/gp_index.html">
   Markovian Gaussian processes
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../ode/ode_index.html">
   Differential equations and SSMs
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../control/control_index.html">
   Optimal control
  </a>
 </li>
 <li class="toctree-l1">
  <a class="reference internal" href="../../bib.html">
   Bibliography
  </a>
 </li>
</ul>

    </div>
</nav> <!-- To handle the deprecated key -->

<div class="navbar_extra_footer">
  Powered by <a href="https://jupyterbook.org">Jupyter Book</a>
</div>

</div>


<main class="col py-md-3 pl-md-4 bd-content overflow-auto" role="main">
    
    <div class="topbar container-xl fixed-top">
    <div class="topbar-contents row">
        <div class="col-12 col-md-3 bd-topbar-whitespace site-navigation show"></div>
        <div class="col pl-md-4 topbar-main">
            
            <button id="navbar-toggler" class="navbar-toggler ml-0" type="button" data-toggle="collapse"
                data-toggle="tooltip" data-placement="bottom" data-target=".site-navigation" aria-controls="navbar-menu"
                aria-expanded="true" aria-label="Toggle navigation" aria-controls="site-navigation"
                title="Toggle navigation" data-toggle="tooltip" data-placement="left">
                <i class="fas fa-bars"></i>
                <i class="fas fa-arrow-left"></i>
                <i class="fas fa-arrow-up"></i>
            </button>
            
            
<div class="dropdown-buttons-trigger">
    <button id="dropdown-buttons-trigger" class="btn btn-secondary topbarbtn" aria-label="Download this page"><i
            class="fas fa-download"></i></button>

    <div class="dropdown-buttons">
        <!-- ipynb file if we had a myst markdown file -->
        
        <!-- Download raw file -->
        <a class="dropdown-buttons" href="../../_sources/chapters/ssm/learning.ipynb"><button type="button"
                class="btn btn-secondary topbarbtn" title="Download source file" data-toggle="tooltip"
                data-placement="left">.ipynb</button></a>
        <!-- Download PDF via print -->
        <button type="button" id="download-print" class="btn btn-secondary topbarbtn" title="Print to PDF"
                onclick="printPdf(this)" data-toggle="tooltip" data-placement="left">.pdf</button>
    </div>
</div>

            <!-- Source interaction buttons -->

<div class="dropdown-buttons-trigger">
    <button id="dropdown-buttons-trigger" class="btn btn-secondary topbarbtn"
        aria-label="Connect with source repository"><i class="fab fa-github"></i></button>
    <div class="dropdown-buttons sourcebuttons">
        <a class="repository-button"
            href="https://github.com/probml/ssm-book"><button type="button" class="btn btn-secondary topbarbtn"
                data-toggle="tooltip" data-placement="left" title="Source repository"><i
                    class="fab fa-github"></i>repository</button></a>
        <a class="issues-button"
            href="https://github.com/probml/ssm-book/issues/new?title=Issue%20on%20page%20%2Fchapters/ssm/learning.html&body=Your%20issue%20content%20here."><button
                type="button" class="btn btn-secondary topbarbtn" data-toggle="tooltip" data-placement="left"
                title="Open an issue"><i class="fas fa-lightbulb"></i>open issue</button></a>
        
    </div>
</div>

            <!-- Full screen (wrap in <a> to have style consistency -->

<a class="full-screen-button"><button type="button" class="btn btn-secondary topbarbtn" data-toggle="tooltip"
        data-placement="bottom" onclick="toggleFullScreen()" aria-label="Fullscreen mode"
        title="Fullscreen mode"><i
            class="fas fa-expand"></i></button></a>

            <!-- Launch buttons -->

<div class="dropdown-buttons-trigger">
    <button id="dropdown-buttons-trigger" class="btn btn-secondary topbarbtn"
        aria-label="Launch interactive content"><i class="fas fa-rocket"></i></button>
    <div class="dropdown-buttons">
        
        <a class="binder-button" href="https://mybinder.org/v2/gh/probml/ssm-book/main?urlpath=tree/chapters/ssm/learning.ipynb"><button type="button"
                class="btn btn-secondary topbarbtn" title="Launch Binder" data-toggle="tooltip"
                data-placement="left"><img class="binder-button-logo"
                    src="../../_static/images/logo_binder.svg"
                    alt="Interact on binder">Binder</button></a>
        
        
        <a class="colab-button" href="https://colab.research.google.com/github/probml/ssm-book/blob/main/chapters/ssm/learning.ipynb"><button type="button" class="btn btn-secondary topbarbtn"
                title="Launch Colab" data-toggle="tooltip" data-placement="left"><img class="colab-button-logo"
                    src="../../_static/images/logo_colab.png"
                    alt="Interact on Colab">Colab</button></a>
        
        
    </div>
</div>

        </div>

        <!-- Table of contents -->
        <div class="d-none d-md-block col-md-2 bd-toc show noprint">
            
        </div>
    </div>
</div>
    <div id="main-content" class="row">
        <div class="col-12 col-md-9 pl-md-3 pr-md-0">
            <!-- Table of contents that is only displayed when printing the page -->
            <div id="jb-print-docs-body" class="onlyprint">
                <h1>Parameter estimation (learning)</h1>
                <!-- Table of contents -->
                <div id="print-main-content">
                    <div id="jb-print-toc">
                        
                    </div>
                </div>
            </div>
            
              <div>
                
  <div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">### Import standard libraries</span>

<span class="kn">import</span> <span class="nn">abc</span>
<span class="kn">from</span> <span class="nn">dataclasses</span> <span class="kn">import</span> <span class="n">dataclass</span>
<span class="kn">import</span> <span class="nn">functools</span>
<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="n">partial</span>
<span class="kn">import</span> <span class="nn">itertools</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Any</span><span class="p">,</span> <span class="n">Callable</span><span class="p">,</span> <span class="n">NamedTuple</span><span class="p">,</span> <span class="n">Optional</span><span class="p">,</span> <span class="n">Union</span><span class="p">,</span> <span class="n">Tuple</span>

<span class="kn">import</span> <span class="nn">jax</span>
<span class="kn">import</span> <span class="nn">jax.numpy</span> <span class="k">as</span> <span class="nn">jnp</span>
<span class="kn">from</span> <span class="nn">jax</span> <span class="kn">import</span> <span class="n">lax</span><span class="p">,</span> <span class="n">vmap</span><span class="p">,</span> <span class="n">jit</span><span class="p">,</span> <span class="n">grad</span>
<span class="c1">#from jax.scipy.special import logit</span>
<span class="c1">#from jax.nn import softmax</span>
<span class="kn">import</span> <span class="nn">jax.random</span> <span class="k">as</span> <span class="nn">jr</span>


<span class="kn">import</span> <span class="nn">distrax</span>
<span class="kn">import</span> <span class="nn">optax</span>

<span class="kn">import</span> <span class="nn">jsl</span>
<span class="kn">import</span> <span class="nn">ssm_jax</span>
</pre></div>
</div>
</div>
</div>
<div class="tex2jax_ignore mathjax_ignore section" id="parameter-estimation-learning">
<span id="sec-learning"></span><h1>Parameter estimation (learning)<a class="headerlink" href="#parameter-estimation-learning" title="Permalink to this headline">¶</a></h1>
<p>So far, we have assumed that the parameters <span class="math notranslate nohighlight">\(\params\)</span> of the SSM are known.
For example, in the case of an HMM with categorical observations
we have <span class="math notranslate nohighlight">\(\params = (\hmmInit, \hmmTrans, \hmmObs)\)</span>,
and in the case of an LDS, we have <span class="math notranslate nohighlight">\(\params = 
(\ldsTrans, \ldsObs, \ldsTransIn, \ldsObsIn, \transCov, \obsCov, \initMean, \initCov)\)</span>.
If we adopt a Bayesian perspective, we can view these parameters as random variables that are
shared across all time steps, and across all sequences.
This is shown in <a class="reference internal" href="#fig-hmm-plates"><span class="std std-numref">Fig. 6</span></a>, where we adopt <span class="math notranslate nohighlight">\(\keyword{plate notation}\)</span>
to represent repetitive structure.</p>
<div class="figure align-default" id="fig-hmm-plates">
<a class="reference internal image-reference" href="../../_images/hmmDgmPlatesY.png"><img alt="../../_images/hmmDgmPlatesY.png" src="../../_images/hmmDgmPlatesY.png" style="width: 285.0px; height: 236.0px;" /></a>
<p class="caption"><span class="caption-number">Fig. 6 </span><span class="caption-text">Illustration of an HMM using plate notation, where we show the parameter
nodes which are shared across all the sequences.</span><a class="headerlink" href="#fig-hmm-plates" title="Permalink to this image">¶</a></p>
</div>
<p>Suppose we observe <span class="math notranslate nohighlight">\(N\)</span> sequences <span class="math notranslate nohighlight">\(\data = \{\obs_{n,1:T_n}: n=1:N\}\)</span>.
Then the goal of <span class="math notranslate nohighlight">\(\keyword{parameter estimation}\)</span>, also called <span class="math notranslate nohighlight">\(\keyword{model learning}\)</span>
or <span class="math notranslate nohighlight">\(\keyword{model fitting}\)</span>, is to approximate the posterior</p>
<div class="amsmath math notranslate nohighlight" id="equation-aeba05bd-181c-4460-a520-00ce9651ff39">
<span class="eqno">(17)<a class="headerlink" href="#equation-aeba05bd-181c-4460-a520-00ce9651ff39" title="Permalink to this equation">¶</a></span>\[\begin{align}
p(\params|\data) \propto p(\params) \prod_{n=1}^N p(\obs_{n,1:T_n} | \params)
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(p(\obs_{n,1:T_n} | \params)\)</span> is the marginal likelihood of sequence <span class="math notranslate nohighlight">\(n\)</span>:</p>
<div class="amsmath math notranslate nohighlight" id="equation-45323cdb-e343-4539-84fc-8bfb3adf2c7e">
<span class="eqno">(18)<a class="headerlink" href="#equation-45323cdb-e343-4539-84fc-8bfb3adf2c7e" title="Permalink to this equation">¶</a></span>\[\begin{align}
p(\obs_{1:T} | \params) = \int  p(\hidden_{1:T}, \obs_{1:T} | \params) d\hidden_{1:T}
\end{align}\]</div>
<p>Since computing the full posterior is computationally difficult, we often settle for computing
a point estimate such as the MAP (maximum a posterior) estimate</p>
<div class="amsmath math notranslate nohighlight" id="equation-430a5016-7826-4b1a-b76a-b25346317ded">
<span class="eqno">(19)<a class="headerlink" href="#equation-430a5016-7826-4b1a-b76a-b25346317ded" title="Permalink to this equation">¶</a></span>\[\begin{align}
\params_{\map} = \arg \max_{\params} \log p(\params) + \sum_{n=1}^N \log p(\obs_{n,1:T_n} | \params)
\end{align}\]</div>
<p>If we ignore the prior term, we get the maximum likelihood estimate or MLE:</p>
<div class="amsmath math notranslate nohighlight" id="equation-466da0d8-afab-49ab-a6ec-f804e2279fb0">
<span class="eqno">(20)<a class="headerlink" href="#equation-466da0d8-afab-49ab-a6ec-f804e2279fb0" title="Permalink to this equation">¶</a></span>\[\begin{align}
\params_{\mle} = \arg \max_{\params}  \sum_{n=1}^N \log p(\obs_{n,1:T_n} | \params)
\end{align}\]</div>
<p>In practice, the MAP estimate often works better than the MLE, since the prior can regularize
the estimate to ensure the model is numerically stable and does not overfit the training set.</p>
<p>We will discuss a variety of algorithms for parameter estimation in later chapters.</p>
</div>

    <script type="text/x-thebe-config">
    {
        requestKernel: true,
        binderOptions: {
            repo: "binder-examples/jupyter-stacks-datascience",
            ref: "master",
        },
        codeMirrorConfig: {
            theme: "abcdef",
            mode: "python"
        },
        kernelOptions: {
            kernelName: "python3",
            path: "./chapters/ssm"
        },
        predefinedOutput: true
    }
    </script>
    <script>kernelName = 'python3'</script>

              </div>
              
            
                <!-- Previous / next buttons -->
<div class='prev-next-area'> 
    <a class='left-prev' id="prev-link" href="inference.html" title="previous page">
        <i class="fas fa-angle-left"></i>
        <div class="prev-next-info">
            <p class="prev-next-subtitle">previous</p>
            <p class="prev-next-title">States estimation (inference)</p>
        </div>
    </a>
    <a class='right-next' id="next-link" href="../hmm/hmm_index.html" title="next page">
    <div class="prev-next-info">
        <p class="prev-next-subtitle">next</p>
        <p class="prev-next-title">Hidden Markov Models</p>
    </div>
    <i class="fas fa-angle-right"></i>
    </a>
</div>
            
        </div>
    </div>
    <footer class="footer">
  <p>
    
      By Kevin Murphy, Scott Linderman, et al.<br/>
    
        &copy; Copyright 2021.<br/>
  </p>
</footer>
</main>


      </div>
    </div>
  
  <script src="../../_static/js/index.be7d3bbb2ef33a8344ce.js"></script>

  </body>
</html>