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   State Space Models: A Modern Approach
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   Sequential Monte Carlo
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<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>
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