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   State Space Models: A Modern Approach
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                <h1>Inferential goals</h1>
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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>
</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="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>
</div>
</div>
</div>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\newcommand\floor[1]{\lfloor#1\rfloor}\\\newcommand{\real}{\mathbb{R}}\\% Numbers
\newcommand{\vzero}{\boldsymbol{0}}
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\newcommand{\vepsilon}{\boldsymbol{\epsilon}}
\newcommand{\vzeta}{\boldsymbol{\zeta}}
\newcommand{\vXi}{\boldsymbol{\Xi}}
\newcommand{\vell}{\boldsymbol{\ell}}
\newcommand{\veta}{\boldsymbol{\eta}}
%\newcommand{\vEta}{\boldsymbol{\Eta}}
\newcommand{\vgamma}{\boldsymbol{\gamma}}
\newcommand{\vGamma}{\boldsymbol{\Gamma}}
\newcommand{\vmu}{\boldsymbol{\mu}}
\newcommand{\vmut}{\boldsymbol{\tilde{\mu}}}
\newcommand{\vnu}{\boldsymbol{\nu}}
\newcommand{\vkappa}{\boldsymbol{\kappa}}
\newcommand{\vlambda}{\boldsymbol{\lambda}}
\newcommand{\vLambda}{\boldsymbol{\Lambda}}
\newcommand{\vLambdaBar}{\overline{\vLambda}}
%\newcommand{\vnu}{\boldsymbol{\nu}}
\newcommand{\vomega}{\boldsymbol{\omega}}
\newcommand{\vOmega}{\boldsymbol{\Omega}}
\newcommand{\vphi}{\boldsymbol{\phi}}
\newcommand{\vvarphi}{\boldsymbol{\varphi}}
\newcommand{\vPhi}{\boldsymbol{\Phi}}
\newcommand{\vpi}{\boldsymbol{\pi}}
\newcommand{\vPi}{\boldsymbol{\Pi}}
\newcommand{\vpsi}{\boldsymbol{\psi}}
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\newcommand{\vthetat}{\boldsymbol{\tilde{\theta}}}
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\newcommand{\vSigma}{\boldsymbol{\Sigma}}
\newcommand{\vSigmat}{\boldsymbol{\tilde{\Sigma}}}
\newcommand{\vsigmoid}{\vsigma}
\newcommand{\vtau}{\boldsymbol{\tau}}
\newcommand{\vxi}{\boldsymbol{\xi}}\\
% Lower Roman (Vectors)
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\newcommand{\vg}{\mathbf{g}}
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%\newcommand{\myvh}{\mathbf{h}}
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\newcommand{\vl}{\mathbf{l}}
\newcommand{\vm}{\mathbf{m}}
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\newcommand{\vp}{\mathbf{p}}
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\newcommand{\vr}{\mathbf{r}}
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\newcommand{\vt}{\mathbf{t}}
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\newcommand{\vw}{\mathbf{w}}
\newcommand{\vws}{\vw_s}
\newcommand{\vwt}{\mathbf{\tilde{w}}}
\newcommand{\vWt}{\mathbf{\tilde{W}}}
\newcommand{\vwh}{\hat{\vw}}
\newcommand{\vx}{\mathbf{x}}
%\newcommand{\vx}{\mathbf{x}}
\newcommand{\vxt}{\mathbf{\tilde{x}}}
\newcommand{\vy}{\mathbf{y}}
\newcommand{\vyt}{\mathbf{\tilde{y}}}
\newcommand{\vz}{\mathbf{z}}
%\newcommand{\vzt}{\mathbf{\tilde{z}}}\\
% Upper Roman (Matrices)
\newcommand{\vA}{\mathbf{A}}
\newcommand{\vB}{\mathbf{B}}
\newcommand{\vC}{\mathbf{C}}
\newcommand{\vD}{\mathbf{D}}
\newcommand{\vE}{\mathbf{E}}
\newcommand{\vF}{\mathbf{F}}
\newcommand{\vG}{\mathbf{G}}
\newcommand{\vH}{\mathbf{H}}
\newcommand{\vI}{\mathbf{I}}
\newcommand{\vJ}{\mathbf{J}}
\newcommand{\vK}{\mathbf{K}}
\newcommand{\vL}{\mathbf{L}}
\newcommand{\vM}{\mathbf{M}}
\newcommand{\vMt}{\mathbf{\tilde{M}}}
\newcommand{\vN}{\mathbf{N}}
\newcommand{\vO}{\mathbf{O}}
\newcommand{\vP}{\mathbf{P}}
\newcommand{\vQ}{\mathbf{Q}}
\newcommand{\vR}{\mathbf{R}}
\newcommand{\vS}{\mathbf{S}}
\newcommand{\vT}{\mathbf{T}}
\newcommand{\vU}{\mathbf{U}}
\newcommand{\vV}{\mathbf{V}}
\newcommand{\vW}{\mathbf{W}}
\newcommand{\vX}{\mathbf{X}}
%\newcommand{\vXs}{\vX_{\vs}}
\newcommand{\vXs}{\vX_{s}}
\newcommand{\vXt}{\mathbf{\tilde{X}}}
\newcommand{\vY}{\mathbf{Y}}
\newcommand{\vZ}{\mathbf{Z}}
\newcommand{\vZt}{\mathbf{\tilde{Z}}}
\newcommand{\vzt}{\mathbf{\tilde{z}}}\\
%%%%
\newcommand{\hidden}{\vz}
\newcommand{\hid}{\hidden}
\newcommand{\observed}{\vy}
\newcommand{\obs}{\observed}
\newcommand{\inputs}{\vu}
\newcommand{\input}{\inputs}\\\newcommand{\hmmTrans}{\vA}
\newcommand{\hmmObs}{\vB}
\newcommand{\hmmInit}{\vpi}
\newcommand{\hmmhid}{\hidden}
\newcommand{\hmmobs}{\obs}\\\newcommand{\ldsDyn}{\vA}
\newcommand{\ldsObs}{\vC}
\newcommand{\ldsDynIn}{\vB}
\newcommand{\ldsObsIn}{\vD}
\newcommand{\ldsDynNoise}{\vQ}
\newcommand{\ldsObsNoise}{\vR}\\\newcommand{\ssmDynFn}{f}
\newcommand{\ssmObsFn}{h}\\
%%%
\newcommand{\gauss}{\mathcal{N}}\\\newcommand{\diag}{\mathrm{diag}}\end{aligned}\end{align} \]</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="fig-dbn-inference">
<a class="reference internal image-reference" href="../../_images/inference-problems-tikz.png"><img alt="../../_images/inference-problems-tikz.png" src="../../_images/inference-problems-tikz.png" style="width: 352.2px; height: 269.4px;" /></a>
<p class="caption"><span class="caption-number">Fig. 7 </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="#fig-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="math notranslate nohighlight">
\[p(\hmmobs_{t+h}|\hmmobs_{1:t})
= \sum_{\hmmhid_{t+h}} p(\hmmobs_{t+h}|\hmmhid_{t+h}) p(\hmmhid_{t+h}|\hmmobs_{1:t})\]</div>
<p>where the hidden state predictive distribution is</p>
<div class="amsmath math notranslate nohighlight" id="equation-37bb812e-8d87-45c3-b352-11b694274a9e">
<span class="eqno">(18)<a class="headerlink" href="#equation-37bb812e-8d87-45c3-b352-11b694274a9e" 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
<a class="reference internal" href="#fig-dbn-inference"><span class="std std-numref">Fig. 7</span></a> 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">
<span id="sec-casino-inference"></span><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> and <span class="xref myst"></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"># 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>

<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="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>
</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>
</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"># 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>
</pre></div>
</div>
</div>
<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;)
</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"># 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>
</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"># 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>
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<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>
 

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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/inference_9_1.png" src="../../_images/inference_9_1.png" />
</div>
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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>


</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;Smoothed&#39;)
</pre></div>
</div>
<img alt="../../_images/inference_10_1.png" src="../../_images/inference_10_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="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Text(0.5, 1.0, &#39;Viterbi&#39;)
</pre></div>
</div>
<img alt="../../_images/inference_11_1.png" src="../../_images/inference_11_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>
<div class="section" id="example-inference-in-the-tracking-lg-ssm">
<h2>Example: inference in the tracking LG-SSM<a class="headerlink" href="#example-inference-in-the-tracking-lg-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="lds.html#sec-tracking-lds"><span class="std std-ref">Example: tracking a 2d point</span></a>.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">314</span><span class="p">)</span>
<span class="n">timesteps</span> <span class="o">=</span> <span class="mi">15</span>
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">jnp</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="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">delta</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="p">])</span>

<span class="n">C</span> <span class="o">=</span> <span class="n">jnp</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="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
<span class="p">])</span>

<span class="n">state_size</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span>
<span class="n">observation_size</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">shape</span>

<span class="n">Q</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">state_size</span><span class="p">)</span> <span class="o">*</span> <span class="mf">0.001</span>
<span class="n">R</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">observation_size</span><span class="p">)</span> <span class="o">*</span> <span class="mf">1.0</span>
<span class="n">mu0</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">8</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
<span class="n">Sigma0</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">state_size</span><span class="p">)</span> <span class="o">*</span> <span class="mf">1.0</span>

<span class="kn">from</span> <span class="nn">jsl.lds.kalman_filter</span> <span class="kn">import</span> <span class="n">LDS</span><span class="p">,</span> <span class="n">smooth</span><span class="p">,</span> <span class="nb">filter</span>

<span class="n">lds</span> <span class="o">=</span> <span class="n">LDS</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">mu0</span><span class="p">,</span> <span class="n">Sigma0</span><span class="p">)</span>


<span class="n">z_hist</span><span class="p">,</span> <span class="n">x_hist</span> <span class="o">=</span> <span class="n">lds</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">timesteps</span><span class="p">)</span>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">jsl.demos.plot_utils</span> <span class="kn">import</span> <span class="n">plot_ellipse</span>

<span class="k">def</span> <span class="nf">plot_tracking_values</span><span class="p">(</span><span class="n">observed</span><span class="p">,</span> <span class="n">filtered</span><span class="p">,</span> <span class="n">cov_hist</span><span class="p">,</span> <span class="n">signal_label</span><span class="p">,</span> <span class="n">ax</span><span class="p">):</span>
    <span class="n">timesteps</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">observed</span><span class="o">.</span><span class="n">shape</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">observed</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">observed</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
            <span class="n">markerfacecolor</span><span class="o">=</span><span class="s2">&quot;none&quot;</span><span class="p">,</span> <span class="n">markeredgewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;observed&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</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="o">*</span><span class="n">filtered</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">signal_label</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;tab:red&quot;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;x&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
    <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">timesteps</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
        <span class="n">covn</span> <span class="o">=</span> <span class="n">cov_hist</span><span class="p">[</span><span class="n">t</span><span class="p">][:</span><span class="mi">2</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span>
        <span class="n">plot_ellipse</span><span class="p">(</span><span class="n">covn</span><span class="p">,</span> <span class="n">filtered</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">],</span> <span class="n">ax</span><span class="p">,</span> <span class="n">n_std</span><span class="o">=</span><span class="mf">2.0</span><span class="p">,</span> <span class="n">plot_center</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">&quot;equal&quot;</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
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</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">mu_hist</span><span class="p">,</span> <span class="n">Sigma_hist</span><span class="p">,</span> <span class="n">mu_cond_hist</span><span class="p">,</span> <span class="n">Sigma_cond_hist</span> <span class="o">=</span> <span class="nb">filter</span><span class="p">(</span><span class="n">lds</span><span class="p">,</span> <span class="n">x_hist</span><span class="p">)</span>
<span class="n">l2_filter</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">z_hist</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="n">mu_hist</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">],</span> <span class="mi">2</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;L2-filter: </span><span class="si">{</span><span class="n">l2_filter</span><span class="si">:</span><span class="s2">0.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="n">fig_filtered</span><span class="p">,</span> <span class="n">axs</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_tracking_values</span><span class="p">(</span><span class="n">x_hist</span><span class="p">,</span> <span class="n">mu_hist</span><span class="p">,</span> <span class="n">Sigma_hist</span><span class="p">,</span> <span class="s2">&quot;filtered&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">)</span>
</pre></div>
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</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>L2-filter: 3.2481
</pre></div>
</div>
<img alt="../../_images/inference_16_1.png" src="../../_images/inference_16_1.png" />
</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"># Smoothing</span>

<span class="n">mu_hist_smooth</span><span class="p">,</span> <span class="n">Sigma_hist_smooth</span> <span class="o">=</span> <span class="n">smooth</span><span class="p">(</span><span class="n">lds</span><span class="p">,</span> <span class="n">mu_hist</span><span class="p">,</span> <span class="n">Sigma_hist</span><span class="p">,</span> <span class="n">mu_cond_hist</span><span class="p">,</span> <span class="n">Sigma_cond_hist</span><span class="p">)</span>
<span class="n">l2_smooth</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">z_hist</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="n">mu_hist_smooth</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">],</span> <span class="mi">2</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;L2-smooth: </span><span class="si">{</span><span class="n">l2_smooth</span><span class="si">:</span><span class="s2">0.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="n">fig_smoothed</span><span class="p">,</span> <span class="n">axs</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_tracking_values</span><span class="p">(</span><span class="n">x_hist</span><span class="p">,</span> <span class="n">mu_hist_smooth</span><span class="p">,</span> <span class="n">Sigma_hist_smooth</span><span class="p">,</span> <span class="s2">&quot;smoothed&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">)</span>
</pre></div>
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</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>L2-smooth: 2.0450
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