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  <a class="reference internal nav-link" href="#">
   What are State Space Models?
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  <a class="reference internal nav-link" href="#hidden-markov-models">
   Hidden Markov Models
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     Example: Casino HMM
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   Linear Gaussian SSMs
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     Example: model for 2d tracking
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   Inferential goals
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     Example: inference in the casino HMM
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                <h1>What are State Space Models?</h1>
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                            <h2> Contents </h2>
                        </div>
                        <nav aria-label="Page">
                            <ul class="visible nav section-nav flex-column">
 <li class="toc-h1 nav-item toc-entry">
  <a class="reference internal nav-link" href="#">
   What are State Space Models?
  </a>
 </li>
 <li class="toc-h1 nav-item toc-entry">
  <a class="reference internal nav-link" href="#hidden-markov-models">
   Hidden Markov Models
  </a>
  <ul class="visible nav section-nav flex-column">
   <li class="toc-h2 nav-item toc-entry">
    <a class="reference internal nav-link" href="#example-casino-hmm">
     Example: Casino HMM
    </a>
   </li>
  </ul>
 </li>
 <li class="toc-h1 nav-item toc-entry">
  <a class="reference internal nav-link" href="#linear-gaussian-ssms">
   Linear Gaussian SSMs
  </a>
  <ul class="visible nav section-nav flex-column">
   <li class="toc-h2 nav-item toc-entry">
    <a class="reference internal nav-link" href="#example-model-for-2d-tracking">
     Example: model for 2d tracking
    </a>
   </li>
  </ul>
 </li>
 <li class="toc-h1 nav-item toc-entry">
  <a class="reference internal nav-link" href="#inferential-goals">
   Inferential goals
  </a>
  <ul class="visible nav section-nav flex-column">
   <li class="toc-h2 nav-item toc-entry">
    <a class="reference internal nav-link" href="#example-inference-in-the-casino-hmm">
     Example: inference in the casino HMM
    </a>
   </li>
   <li class="toc-h2 nav-item toc-entry">
    <a class="reference internal nav-link" href="#example-inference-in-the-tracking-ssm">
     Example: inference in the tracking SSM
    </a>
   </li>
  </ul>
 </li>
</ul>

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              <div>
                
  <div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># meta-data does not work yet in VScode</span>
<span class="c1"># https://github.com/microsoft/vscode-jupyter/issues/1121</span>

<span class="p">{</span>
    <span class="s2">&quot;tags&quot;</span><span class="p">:</span> <span class="p">[</span>
        <span class="s2">&quot;hide-cell&quot;</span>
    <span class="p">]</span>
<span class="p">}</span>


<span class="c1">### Install necessary libraries</span>

<span class="k">try</span><span class="p">:</span>
    <span class="kn">import</span> <span class="nn">jax</span>
<span class="k">except</span><span class="p">:</span>
    <span class="c1"># For cuda version, see https://github.com/google/jax#installation</span>
    <span class="o">%</span><span class="k">pip</span> install --upgrade &quot;jax[cpu]&quot; 
    <span class="kn">import</span> <span class="nn">jax</span>

<span class="k">try</span><span class="p">:</span>
    <span class="kn">import</span> <span class="nn">distrax</span>
<span class="k">except</span><span class="p">:</span>
    <span class="o">%</span><span class="k">pip</span> install --upgrade  distrax
    <span class="kn">import</span> <span class="nn">distrax</span>

<span class="k">try</span><span class="p">:</span>
    <span class="kn">import</span> <span class="nn">jsl</span>
<span class="k">except</span><span class="p">:</span>
    <span class="o">%</span><span class="k">pip</span> install git+https://github.com/probml/jsl
    <span class="kn">import</span> <span class="nn">jsl</span>

<span class="k">try</span><span class="p">:</span>
    <span class="kn">import</span> <span class="nn">rich</span>
<span class="k">except</span><span class="p">:</span>
    <span class="o">%</span><span class="k">pip</span> install rich
    <span class="kn">import</span> <span class="nn">rich</span>
</pre></div>
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<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="p">{</span>
    <span class="s2">&quot;tags&quot;</span><span class="p">:</span> <span class="p">[</span>
        <span class="s2">&quot;hide-cell&quot;</span>
    <span class="p">]</span>
<span class="p">}</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">import</span> <span class="nn">itertools</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">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">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="kn">from</span> <span class="nn">jax.scipy.special</span> <span class="kn">import</span> <span class="n">logit</span>
<span class="kn">from</span> <span class="nn">jax.nn</span> <span class="kn">import</span> <span class="n">softmax</span>
<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="n">partial</span>
<span class="kn">from</span> <span class="nn">jax.random</span> <span class="kn">import</span> <span class="n">PRNGKey</span><span class="p">,</span> <span class="n">split</span>

<span class="kn">import</span> <span class="nn">inspect</span>
<span class="kn">import</span> <span class="nn">inspect</span> <span class="k">as</span> <span class="nn">py_inspect</span>
<span class="kn">import</span> <span class="nn">rich</span>
<span class="kn">from</span> <span class="nn">rich</span> <span class="kn">import</span> <span class="n">inspect</span> <span class="k">as</span> <span class="n">r_inspect</span>
<span class="kn">from</span> <span class="nn">rich</span> <span class="kn">import</span> <span class="nb">print</span> <span class="k">as</span> <span class="n">r_print</span>

<span class="k">def</span> <span class="nf">print_source</span><span class="p">(</span><span class="n">fname</span><span class="p">):</span>
    <span class="n">r_print</span><span class="p">(</span><span class="n">py_inspect</span><span class="o">.</span><span class="n">getsource</span><span class="p">(</span><span class="n">fname</span><span class="p">))</span>
</pre></div>
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<div class="math notranslate nohighlight">
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<div class="tex2jax_ignore mathjax_ignore section" id="what-are-state-space-models">
<span id="sec-ssm-intro"></span><h1>What are State Space Models?<a class="headerlink" href="#what-are-state-space-models" title="Permalink to this headline">¶</a></h1>
<p>A state space model or SSM
is a partially observed Markov model,
in which the hidden state,  <span class="math notranslate nohighlight">\(\hidden_t\)</span>,
evolves over time according to a Markov process,
possibly conditional on external inputs or controls <span class="math notranslate nohighlight">\(\input_t\)</span>,
and each hidden state generates some
observations <span class="math notranslate nohighlight">\(\obs_t\)</span> at each time step.
(In this book, we mostly focus on discrete time systems,
although  we consider the continuous-time case in  XXX.)
We get to see the observations, but not the hidden state.
Our main goal is to infer the hidden state given the observations.
However, we can also use the model to predict future observations,
by first predicting future hidden states, and then predicting
what observations they might generate.
By using  a hidden state <span class="math notranslate nohighlight">\(\hidden_t\)</span>
to represent the past observations, <span class="math notranslate nohighlight">\(\obs_{1:t-1}\)</span>,
the  model can have ``infinite’’ memory,
unlike a standard Markov model.</p>
<p>Formally we can define an SSM
as the following joint distribution:</p>
<div class="math notranslate nohighlight" id="equation-ssmfull">
<span class="eqno">(2)<a class="headerlink" href="#equation-ssmfull" title="Permalink to this equation">¶</a></span>\[p(\hmmobs_{1:T},\hmmhid_{1:T}|\inputs_{1:T})
 = \left[ p(\hmmhid_1|\inputs_1) \prod_{t=2}^{T}
 p(\hmmhid_t|\hmmhid_{t-1},\inputs_t) \right]
 \left[ \prod_{t=1}^T p(\hmmobs_t|\hmmhid_t, \inputs_t, \hmmobs_{t-1}) \right]\]</div>
<p>where <span class="math notranslate nohighlight">\(p(\hmmhid_t|\hmmhid_{t-1},\inputs_t)\)</span> is the
transition model,
<span class="math notranslate nohighlight">\(p(\hmmobs_t|\hmmhid_t, \inputs_t, \hmmobs_{t-1})\)</span> is the
observation model,
and <span class="math notranslate nohighlight">\(\inputs_{t}\)</span> is an optional input or action.
See <a class="reference internal" href="#ssm-ar"><span class="std std-numref">Figure 3</span></a>
for an illustration of the corresponding graphical model.</p>
<div class="figure align-default" id="ssm-ar">
<a class="reference internal image-reference" href="../../_images/SSM-AR-inputs.png"><img alt="../../_images/SSM-AR-inputs.png" src="../../_images/SSM-AR-inputs.png" style="width: 152.0px; height: 165.0px;" /></a>
<p class="caption"><span class="caption-number">Fig. 3 </span><span class="caption-text">Illustration of an SSM as a graphical model.</span><a class="headerlink" href="#ssm-ar" title="Permalink to this image">¶</a></p>
</div>
<p>We often consider a simpler setting in which there
are no external inputs,
and the observations are conditionally independent of each other
(rather than having Markovian dependencies) given the hidden state.
In this case the joint simplifies to</p>
<div class="math notranslate nohighlight" id="equation-ssmsimplified">
<span class="eqno">(3)<a class="headerlink" href="#equation-ssmsimplified" title="Permalink to this equation">¶</a></span>\[p(\hmmobs_{1:T},\hmmhid_{1:T})
 = \left[ p(\hmmhid_1) \prod_{t=2}^{T}
 p(\hmmhid_t|\hmmhid_{t-1}) \right]
 \left[ \prod_{t=1}^T p(\hmmobs_t|\hmmhid_t \right]\]</div>
<p>See <a class="reference internal" href="#ssm-simplified"><span class="std std-numref">Figure 4</span></a>
for an illustration of the corresponding graphical model.
Compare <a class="reference internal" href="#equation-ssmfull">(2)</a> and <a class="reference internal" href="#equation-ssmsimplified">(3)</a>.</p>
<div class="figure align-default" id="ssm-simplified">
<a class="reference internal image-reference" href="../../_images/SSM-simplified.png"><img alt="../../_images/SSM-simplified.png" src="../../_images/SSM-simplified.png" style="width: 136.0px; height: 98.0px;" /></a>
<p class="caption"><span class="caption-number">Fig. 4 </span><span class="caption-text">Illustration of a simplified SSM.</span><a class="headerlink" href="#ssm-simplified" title="Permalink to this image">¶</a></p>
</div>
</div>
<div class="tex2jax_ignore mathjax_ignore section" id="hidden-markov-models">
<span id="sec-hmm-intro"></span><h1>Hidden Markov Models<a class="headerlink" href="#hidden-markov-models" title="Permalink to this headline">¶</a></h1>
<p>In this section, we discuss the
hidden Markov model or HMM,
which is a state space model in which the hidden states
are discrete, so <span class="math notranslate nohighlight">\(\hmmhid_t \in \{1,\ldots, K\}\)</span>.
The observations may be discrete,
<span class="math notranslate nohighlight">\(\hmmobs_t \in \{1,\ldots, C\}\)</span>,
or continuous,
<span class="math notranslate nohighlight">\(\hmmobs_t \in \real^D\)</span>,
or some combination,
as we illustrate below.
More details can be found in e.g.,
<span id="id1">[<a class="reference internal" href="../../bib.html#id34" title="O. Cappe, E. Moulines, and T. Ryden. Inference in Hidden Markov Models. Springer, 2005.">CMR05</a>, <a class="reference internal" href="../../bib.html#id33" title="A. Fraser. Hidden Markov Models and Dynamical Systems. SIAM Press, 2008.">Fra08</a>, <a class="reference internal" href="../../bib.html#id32" title="L. R. Rabiner. A tutorial on Hidden Markov Models and selected applications in speech recognition. Proc. of the IEEE, 77(2):257–286, 1989.">Rab89</a>]</span>.
For an interactive introduction,
see <a class="reference external" href="https://nipunbatra.github.io/hmm/">https://nipunbatra.github.io/hmm/</a>.</p>
<div class="section" id="example-casino-hmm">
<h2>Example: Casino HMM<a class="headerlink" href="#example-casino-hmm" title="Permalink to this headline">¶</a></h2>
<p>To illustrate HMMs with categorical observation model,
we consider the “Ocassionally dishonest casino” model from <span id="id2">[<a class="reference internal" href="../../bib.html#id3" title="R. Durbin, S. Eddy, A. Krogh, and G. Mitchison. Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press, 1998.">DEKM98</a>]</span>.
There are 2 hidden states, representing whether the dice being used in the casino is fair or loaded.
Each state defines a distribution over the 6 possible observations.</p>
<p>The transition model is denoted by</p>
<div class="math notranslate nohighlight">
\[p(z_t=j|z_{t-1}=i) = \hmmTrans_{ij}\]</div>
<p>Here the <span class="math notranslate nohighlight">\(i\)</span>’th row of <span class="math notranslate nohighlight">\(\vA\)</span> corresponds to the outgoing distribution from state <span class="math notranslate nohighlight">\(i\)</span>.
This is  a row stochastic matrix,
meaning each row sums to one.
We can visualize
the non-zero entries in the transition matrix by creating a state transition diagram,
as shown in
<a class="reference internal" href="#casino-fig"><span class="std std-numref">Figure 5</span></a></p>
<div class="figure align-default" id="casino-fig">
<a class="reference internal image-reference" href="../../_images/casino.png"><img alt="../../_images/casino.png" src="../../_images/casino.png" style="width: 208.5px; height: 142.5px;" /></a>
<p class="caption"><span class="caption-number">Fig. 5 </span><span class="caption-text">Illustration of the casino HMM.</span><a class="headerlink" href="#casino-fig" title="Permalink to this image">¶</a></p>
</div>
<p>The  observation model
<span class="math notranslate nohighlight">\(p(\obs_t|\hidden_t=j)\)</span> has the form</p>
<div class="math notranslate nohighlight">
\[p(\obs_t=k|\hidden_t=j) = \hmmObs_{jk} \]</div>
<p>This is represented by the histograms associated with each
state in <a class="reference internal" href="#casino-fig"><span class="std std-ref">Illustration of the casino HMM.</span></a>.</p>
<p>Finally,
the initial state distribution is denoted by</p>
<div class="math notranslate nohighlight">
\[p(z_1=j) = \hmmInit_j\]</div>
<p>Collectively we denote all the parameters by <span class="math notranslate nohighlight">\(\vtheta=(\hmmTrans, \hmmObs, \hmmInit)\)</span>.</p>
<p>Now let us implement this model in code.</p>
<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"># state transition matrix</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span>
    <span class="p">[</span><span class="mf">0.95</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">],</span>
    <span class="p">[</span><span class="mf">0.10</span><span class="p">,</span> <span class="mf">0.90</span><span class="p">]</span>
<span class="p">])</span>

<span class="c1"># observation matrix</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span>
    <span class="p">[</span><span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">],</span> <span class="c1"># fair die</span>
    <span class="p">[</span><span class="mi">1</span><span class="o">/</span><span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="o">/</span><span class="mi">10</span><span class="p">]</span> <span class="c1"># loaded die</span>
<span class="p">])</span>

<span class="n">pi</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">])</span>

<span class="p">(</span><span class="n">nstates</span><span class="p">,</span> <span class="n">nobs</span><span class="p">)</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
</pre></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="kn">import</span> <span class="nn">distrax</span>
<span class="kn">from</span> <span class="nn">distrax</span> <span class="kn">import</span> <span class="n">HMM</span>


<span class="n">hmm</span> <span class="o">=</span> <span class="n">HMM</span><span class="p">(</span><span class="n">trans_dist</span><span class="o">=</span><span class="n">distrax</span><span class="o">.</span><span class="n">Categorical</span><span class="p">(</span><span class="n">probs</span><span class="o">=</span><span class="n">A</span><span class="p">),</span>
            <span class="n">init_dist</span><span class="o">=</span><span class="n">distrax</span><span class="o">.</span><span class="n">Categorical</span><span class="p">(</span><span class="n">probs</span><span class="o">=</span><span class="n">pi</span><span class="p">),</span>
            <span class="n">obs_dist</span><span class="o">=</span><span class="n">distrax</span><span class="o">.</span><span class="n">Categorical</span><span class="p">(</span><span class="n">probs</span><span class="o">=</span><span class="n">B</span><span class="p">))</span>

<span class="nb">print</span><span class="p">(</span><span class="n">hmm</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;distrax._src.utils.hmm.HMM object at 0x7fb570b7e1f0&gt;
</pre></div>
</div>
</div>
</div>
<p>Let’s sample from the model. We will generate a sequence of latent states, <span class="math notranslate nohighlight">\(\hid_{1:T}\)</span>,
which we then convert to a sequence of observations, <span class="math notranslate nohighlight">\(\obs_{1:T}\)</span>.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">seed</span> <span class="o">=</span> <span class="mi">314</span>
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">300</span>
<span class="n">z_hist</span><span class="p">,</span> <span class="n">x_hist</span> <span class="o">=</span> <span class="n">hmm</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="n">PRNGKey</span><span class="p">(</span><span class="n">seed</span><span class="p">),</span> <span class="n">seq_len</span><span class="o">=</span><span class="n">n_samples</span><span class="p">)</span>

<span class="n">z_hist_str</span> <span class="o">=</span> <span class="s2">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">z_hist</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">str</span><span class="p">))[:</span><span class="mi">60</span><span class="p">]</span>
<span class="n">x_hist_str</span> <span class="o">=</span> <span class="s2">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">x_hist</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">str</span><span class="p">))[:</span><span class="mi">60</span><span class="p">]</span>

<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Printing sample observed/latent...&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x: </span><span class="si">{</span><span class="n">x_hist_str</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;z: </span><span class="si">{</span><span class="n">z_hist_str</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</pre></div>
</div>
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<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Printing sample observed/latent...
x: 633665342652353616444236412331351246651613325161656366246242
z: 222222211111111111111111111111111111111222111111112222211111
</pre></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"># Here is the source code for the sampling algorithm.</span>

<span class="n">print_source</span><span class="p">(</span><span class="n">hmm</span><span class="o">.</span><span class="n">sample</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_html"><pre style="white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace">  def sample<span style="font-weight: bold">(</span>self,
             *,
             seed: chex.PRNGKey,
             seq_len: chex.Array<span style="font-weight: bold">)</span> -&gt; Tuple:
    <span style="color: #008000; text-decoration-color: #008000">""</span>"Sample from this HMM.

    Samples an observation of given length according to this
    Hidden Markov Model and gives the sequence of the hidden states
    as well as the observation.

    Args:
      seed: Random key of shape <span style="font-weight: bold">(</span><span style="color: #000080; text-decoration-color: #000080; font-weight: bold">2</span>,<span style="font-weight: bold">)</span> and dtype uint32.
      seq_len: The length of the observation sequence.

    Returns:
      Tuple of hidden state sequence, and observation sequence.
    <span style="color: #008000; text-decoration-color: #008000">""</span>"
    rng_key, rng_init = jax.random.split<span style="font-weight: bold">(</span>seed<span style="font-weight: bold">)</span>
    initial_state = self._init_dist.sample<span style="font-weight: bold">(</span><span style="color: #808000; text-decoration-color: #808000">seed</span>=<span style="color: #800080; text-decoration-color: #800080">rng_init</span><span style="font-weight: bold">)</span>

    def draw_state<span style="font-weight: bold">(</span>prev_state, key<span style="font-weight: bold">)</span>:
      state = self._trans_dist.sample<span style="font-weight: bold">(</span><span style="color: #808000; text-decoration-color: #808000">seed</span>=<span style="color: #800080; text-decoration-color: #800080">key</span><span style="font-weight: bold">)</span>
      return state, state

    rng_state, rng_obs = jax.random.split<span style="font-weight: bold">(</span>rng_key<span style="font-weight: bold">)</span>
    keys = jax.random.split<span style="font-weight: bold">(</span>rng_state, seq_len - <span style="color: #000080; text-decoration-color: #000080; font-weight: bold">1</span><span style="font-weight: bold">)</span>
    _, states = jax.lax.scan<span style="font-weight: bold">(</span>draw_state, initial_state, keys<span style="font-weight: bold">)</span>
    states = jnp.append<span style="font-weight: bold">(</span>initial_state, states<span style="font-weight: bold">)</span>

    def draw_obs<span style="font-weight: bold">(</span>state, key<span style="font-weight: bold">)</span>:
      return self._obs_dist.sample<span style="font-weight: bold">(</span><span style="color: #808000; text-decoration-color: #808000">seed</span>=<span style="color: #800080; text-decoration-color: #800080">key</span><span style="font-weight: bold">)</span>

    keys = jax.random.split<span style="font-weight: bold">(</span>rng_obs, seq_len<span style="font-weight: bold">)</span>
    obs_seq = jax.vmap<span style="font-weight: bold">(</span>draw_obs, <span style="color: #808000; text-decoration-color: #808000">in_axes</span>=<span style="font-weight: bold">(</span><span style="color: #000080; text-decoration-color: #000080; font-weight: bold">0</span>, <span style="color: #000080; text-decoration-color: #000080; font-weight: bold">0</span><span style="font-weight: bold">))(</span>states, keys<span style="font-weight: bold">)</span>

    return states, obs_seq

</pre>
</div></div>
</div>
<p>Our primary goal will be to infer the latent state from the observations,
so we can detect if the casino is being dishonest or not. This will
affect how we choose to gamble our money.
We discuss various ways to perform this inference below.</p>
</div>
</div>
<div class="tex2jax_ignore mathjax_ignore section" id="linear-gaussian-ssms">
<h1>Linear Gaussian SSMs<a class="headerlink" href="#linear-gaussian-ssms" title="Permalink to this headline">¶</a></h1>
<p>Blah blah</p>
<div class="section" id="example-model-for-2d-tracking">
<span id="sec-tracking-lds"></span><h2>Example: model for 2d tracking<a class="headerlink" href="#example-model-for-2d-tracking" title="Permalink to this headline">¶</a></h2>
<p>Blah blah</p>
</div>
</div>
<div class="tex2jax_ignore mathjax_ignore section" id="inferential-goals">
<span id="sec-inference"></span><h1>Inferential goals<a class="headerlink" href="#inferential-goals" title="Permalink to this headline">¶</a></h1>
<div class="figure align-default" id="dbn-inference">
<a class="reference internal image-reference" href="../../_images/dbn-inference-problems.png"><img alt="../../_images/dbn-inference-problems.png" src="../../_images/dbn-inference-problems.png" style="width: 512.0px; height: 359.0px;" /></a>
<p class="caption"><span class="caption-number">Fig. 6 </span><span class="caption-text">Illustration of the different kinds of inference in an SSM.
The main kinds of inference for state-space models.
The shaded region is the interval for which we have data.
The arrow represents the time step at which we want to perform inference.
<span class="math notranslate nohighlight">\(t\)</span> is the current time,  <span class="math notranslate nohighlight">\(T\)</span> is the sequence length,
<span class="math notranslate nohighlight">\(\ell\)</span> is the lag and <span class="math notranslate nohighlight">\(h\)</span> is the prediction horizon.</span><a class="headerlink" href="#dbn-inference" title="Permalink to this image">¶</a></p>
</div>
<p>Given the sequence of observations, and a known model,
one of the main tasks with SSMs
to perform posterior inference,
about the hidden states; this is also called
state estimation.
At each time step <span class="math notranslate nohighlight">\(t\)</span>,
there are multiple forms of posterior we may be interested in computing,
including the following:</p>
<ul class="simple">
<li><p>the filtering distribution
<span class="math notranslate nohighlight">\(p(\hmmhid_t|\hmmobs_{1:t})\)</span></p></li>
<li><p>the smoothing distribution
<span class="math notranslate nohighlight">\(p(\hmmhid_t|\hmmobs_{1:T})\)</span> (note that this conditions on future data <span class="math notranslate nohighlight">\(T&gt;t\)</span>)</p></li>
<li><p>the fixed-lag smoothing distribution
<span class="math notranslate nohighlight">\(p(\hmmhid_{t-\ell}|\hmmobs_{1:t})\)</span> (note that this
infers <span class="math notranslate nohighlight">\(\ell\)</span> steps in the past given data up to the present).</p></li>
</ul>
<p>We may also want to compute the
predictive distribution <span class="math notranslate nohighlight">\(h\)</span> steps into the future:</p>
<div class="amsmath math notranslate nohighlight" id="equation-81c90c98-dd2c-4cd3-a4ef-b79cf155d9bf">
<span class="eqno">(4)<a class="headerlink" href="#equation-81c90c98-dd2c-4cd3-a4ef-b79cf155d9bf" title="Permalink to this equation">¶</a></span>\[\begin{align}
p(\hmmobs_{t+h}|\hmmobs_{1:t})
&amp;= \sum_{\hmmhid_{t+h}} p(\hmmobs_{t+h}|\hmmhid_{t+h}) p(\hmmhid_{t+h}|\hmmobs_{1:t})
\end{align}\]</div>
<p>where the hidden state predictive distribution is</p>
<div class="amsmath math notranslate nohighlight" id="equation-ce97a09c-fef8-49e4-ba8b-cacf7c235798">
<span class="eqno">(5)<a class="headerlink" href="#equation-ce97a09c-fef8-49e4-ba8b-cacf7c235798" title="Permalink to this equation">¶</a></span>\[\begin{align}
p(\hmmhid_{t+h}|\hmmobs_{1:t})
&amp;= \sum_{\hmmhid_{t:t+h-1}}
 p(\hmmhid_t|\hmmobs_{1:t}) 
 p(\hmmhid_{t+1}|\hmmhid_{t})
 p(\hmmhid_{t+2}|\hmmhid_{t+1})
\cdots
 p(\hmmhid_{t+h}|\hmmhid_{t+h-1})
\end{align}\]</div>
<p>See \cref{fig:dbn_inf_problems} for a summary of these distributions.</p>
<p>In addition  to comuting posterior marginals,
we may want to compute the most probable hidden sequence,
i.e., the joint MAP estimate</p>
<div class="math notranslate nohighlight">
\[\arg \max_{\hmmhid_{1:T}} p(\hmmhid_{1:T}|\hmmobs_{1:T})\]</div>
<p>or sample sequences from the posterior</p>
<div class="math notranslate nohighlight">
\[\hmmhid_{1:T} \sim p(\hmmhid_{1:T}|\hmmobs_{1:T})\]</div>
<p>Algorithms for all these task are discussed in the following chapters,
since the details depend on the form of the SSM.</p>
<div class="section" id="example-inference-in-the-casino-hmm">
<h2>Example: inference in the casino HMM<a class="headerlink" href="#example-inference-in-the-casino-hmm" title="Permalink to this headline">¶</a></h2>
<p>We now illustrate filtering, smoothing and MAP decoding applied
to the casino HMM from <span class="xref std std-ref">sec:casino</span>.</p>
<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"># Call inference engine</span>

<span class="n">filtered_dist</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">smoothed_dist</span><span class="p">,</span> <span class="n">loglik</span> <span class="o">=</span> <span class="n">hmm</span><span class="o">.</span><span class="n">forward_backward</span><span class="p">(</span><span class="n">x_hist</span><span class="p">)</span>
<span class="n">map_path</span> <span class="o">=</span> <span class="n">hmm</span><span class="o">.</span><span class="n">viterbi</span><span class="p">(</span><span class="n">x_hist</span><span class="p">)</span>
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<div class="cell_output docutils container">
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/anaconda3/lib/python3.8/site-packages/jax/_src/numpy/lax_numpy.py:5154: UserWarning: Explicitly requested dtype &lt;class &#39;jax._src.numpy.lax_numpy.int64&#39;&gt; requested in astype is not available, and will be truncated to dtype int32. To enable more dtypes, set the jax_enable_x64 configuration option or the JAX_ENABLE_X64 shell environment variable. See https://github.com/google/jax#current-gotchas for more.
  lax_internal._check_user_dtype_supported(dtype, &quot;astype&quot;)
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</div>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Find the span of timesteps that the    simulated systems turns to be in state 1</span>
<span class="k">def</span> <span class="nf">find_dishonest_intervals</span><span class="p">(</span><span class="n">z_hist</span><span class="p">):</span>
    <span class="n">spans</span> <span class="o">=</span> <span class="p">[]</span>
    <span class="n">x_init</span> <span class="o">=</span> <span class="mi">0</span>
    <span class="k">for</span> <span class="n">t</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">z_hist</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
        <span class="k">if</span> <span class="n">z_hist</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">z_hist</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
            <span class="n">x_end</span> <span class="o">=</span> <span class="n">t</span>
            <span class="n">spans</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">x_init</span><span class="p">,</span> <span class="n">x_end</span><span class="p">))</span>
        <span class="k">elif</span> <span class="n">z_hist</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">z_hist</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
            <span class="n">x_init</span> <span class="o">=</span> <span class="n">t</span> <span class="o">+</span> <span class="mi">1</span>
    <span class="k">return</span> <span class="n">spans</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot posterior</span>
<span class="k">def</span> <span class="nf">plot_inference</span><span class="p">(</span><span class="n">inference_values</span><span class="p">,</span> <span class="n">z_hist</span><span class="p">,</span> <span class="n">ax</span><span class="p">,</span> <span class="n">state</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">map_estimate</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
    <span class="n">n_samples</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">inference_values</span><span class="p">)</span>
    <span class="n">xspan</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_samples</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
    <span class="n">spans</span> <span class="o">=</span> <span class="n">find_dishonest_intervals</span><span class="p">(</span><span class="n">z_hist</span><span class="p">)</span>
    <span class="k">if</span> <span class="n">map_estimate</span><span class="p">:</span>
        <span class="n">ax</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">xspan</span><span class="p">,</span> <span class="n">inference_values</span><span class="p">,</span> <span class="n">where</span><span class="o">=</span><span class="s2">&quot;post&quot;</span><span class="p">)</span>
    <span class="k">else</span><span class="p">:</span>
        <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xspan</span><span class="p">,</span> <span class="n">inference_values</span><span class="p">[:,</span> <span class="n">state</span><span class="p">])</span>

    <span class="k">for</span> <span class="n">span</span> <span class="ow">in</span> <span class="n">spans</span><span class="p">:</span>
        <span class="n">ax</span><span class="o">.</span><span class="n">axvspan</span><span class="p">(</span><span class="o">*</span><span class="n">span</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">facecolor</span><span class="o">=</span><span class="s2">&quot;tab:gray&quot;</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s2">&quot;none&quot;</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_samples</span><span class="p">)</span>
    <span class="c1"># ax.set_ylim(0, 1)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Observation number&quot;</span><span class="p">)</span>
</pre></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"># Filtering</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">plot_inference</span><span class="p">(</span><span class="n">filtered_dist</span><span class="p">,</span> <span class="n">z_hist</span><span class="p">,</span> <span class="n">ax</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;p(loaded)&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Filtered&quot;</span><span class="p">)</span>
 

</pre></div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Text(0.5, 1.0, &#39;Filtered&#39;)
</pre></div>
</div>
<img alt="../../_images/ssm_19_1.png" src="../../_images/ssm_19_1.png" />
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Smoothing</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">plot_inference</span><span class="p">(</span><span class="n">smoothed_dist</span><span class="p">,</span> <span class="n">z_hist</span><span class="p">,</span> <span class="n">ax</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;p(loaded)&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Smoothed&quot;</span><span class="p">)</span>


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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Text(0.5, 1.0, &#39;Smoothed&#39;)
</pre></div>
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<img alt="../../_images/ssm_20_1.png" src="../../_images/ssm_20_1.png" />
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># MAP estimation</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">plot_inference</span><span class="p">(</span><span class="n">map_path</span><span class="p">,</span> <span class="n">z_hist</span><span class="p">,</span> <span class="n">ax</span><span class="p">,</span> <span class="n">map_estimate</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;MAP state&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Viterbi&quot;</span><span class="p">)</span>
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<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Text(0.5, 1.0, &#39;Viterbi&#39;)
</pre></div>
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<img alt="../../_images/ssm_21_1.png" src="../../_images/ssm_21_1.png" />
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># TODO: posterior samples</span>
</pre></div>
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<div class="section" id="example-inference-in-the-tracking-ssm">
<h2>Example: inference in the tracking SSM<a class="headerlink" href="#example-inference-in-the-tracking-ssm" title="Permalink to this headline">¶</a></h2>
<p>We now illustrate filtering, smoothing and MAP decoding applied
to the 2d tracking HMM from <a class="reference internal" href="#sec-tracking-lds"><span class="std std-ref">Example: model for 2d tracking</span></a>.</p>
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