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   State Space Models: A Modern Approach
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                <h1>Hidden Markov Models</h1>
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                        </nav>
                    </div>
                </div>
            </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">\(\hidden_t \in \{1,\ldots, \nstates\}\)</span>.
The observations may be discrete,
<span class="math notranslate nohighlight">\(\obs_t \in \{1,\ldots, \nsymbols\}\)</span>,
or continuous,
<span class="math notranslate nohighlight">\(\obs_t \in \real^\nstates\)</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="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">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>

<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="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">
\[p(\hidden_t=j|\hidden_{t-1}=i) = \hmmTransScalar_{ij}\]</div>
<p>Here the <span class="math notranslate nohighlight">\(i\)</span>’th row of <span class="math notranslate nohighlight">\(\hmmTrans\)</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="#fig-casino"><span class="std std-numref">Fig. 3</span></a>.</p>
<div class="figure align-default" id="fig-casino">
<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. 3 </span><span class="caption-text">Illustration of the casino HMM.</span><a class="headerlink" href="#fig-casino" 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">
\[p(\obs_t=k|\hidden_t=j) = \hmmObsScalar_{jk} \]</div>
<p>This is represented by the histograms associated with each
state in  <a class="reference internal" href="#fig-casino"><span class="std std-numref">Fig. 3</span></a>.</p>
<p>Finally,
the initial state distribution is denoted by</p>
<div class="math notranslate nohighlight">
\[p(\hidden_1=j) = \hmmInitScalar_j\]</div>
<p>Collectively we denote all the parameters by <span class="math notranslate nohighlight">\(\params=(\hmmTrans, \hmmObs, \hmmInit)\)</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 0x7fbca906ad60&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">\(\hidden_{1:T}\)</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">jr</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>
</div>
<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>
<p>Below is the source code for the sampling algorithm.</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">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>Let us check correctness by computing empirical pairwise statistics</p>
<p>We will compute the number of i-&gt;j latent state transitions, and check that it is close to the true
A[i,j] transition probabilites.</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="kn">import</span> <span class="nn">collections</span>
<span class="k">def</span> <span class="nf">compute_counts</span><span class="p">(</span><span class="n">state_seq</span><span class="p">,</span> <span class="n">nstates</span><span class="p">):</span>
    <span class="n">wseq</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="n">state_seq</span><span class="p">)</span>
    <span class="n">word_pairs</span> <span class="o">=</span> <span class="p">[</span><span class="n">pair</span> <span class="k">for</span> <span class="n">pair</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">wseq</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">wseq</span><span class="p">[</span><span class="mi">1</span><span class="p">:])]</span>
    <span class="n">counter_pairs</span> <span class="o">=</span> <span class="n">collections</span><span class="o">.</span><span class="n">Counter</span><span class="p">(</span><span class="n">word_pairs</span><span class="p">)</span>
    <span class="n">counts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nstates</span><span class="p">,</span> <span class="n">nstates</span><span class="p">))</span>
    <span class="k">for</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">v</span><span class="p">)</span> <span class="ow">in</span> <span class="n">counter_pairs</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
        <span class="n">counts</span><span class="p">[</span><span class="n">k</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">k</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span> <span class="o">=</span> <span class="n">v</span>
    <span class="k">return</span> <span class="n">counts</span>


<span class="k">def</span> <span class="nf">normalize</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">1e-15</span><span class="p">):</span>
    <span class="n">u</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">u</span> <span class="o">==</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">jnp</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">u</span> <span class="o">&lt;</span> <span class="n">eps</span><span class="p">,</span> <span class="n">eps</span><span class="p">,</span> <span class="n">u</span><span class="p">))</span>
    <span class="n">c</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="n">axis</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">where</span><span class="p">(</span><span class="n">c</span> <span class="o">==</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">u</span> <span class="o">/</span> <span class="n">c</span><span class="p">,</span> <span class="n">c</span>

<span class="k">def</span> <span class="nf">normalize_counts</span><span class="p">(</span><span class="n">counts</span><span class="p">):</span>
    <span class="n">ncounts</span> <span class="o">=</span> <span class="n">vmap</span><span class="p">(</span><span class="k">lambda</span> <span class="n">v</span><span class="p">:</span> <span class="n">normalize</span><span class="p">(</span><span class="n">v</span><span class="p">)[</span><span class="mi">0</span><span class="p">],</span> <span class="n">in_axes</span><span class="o">=</span><span class="mi">0</span><span class="p">)(</span><span class="n">counts</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">ncounts</span>

<span class="n">init_dist</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="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
<span class="n">trans_mat</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="mf">0.7</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</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="n">obs_mat</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="mi">2</span><span class="p">)</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">trans_mat</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">init_dist</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">obs_mat</span><span class="p">))</span>

<span class="n">rng_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">0</span><span class="p">)</span>
<span class="n">seq_len</span> <span class="o">=</span> <span class="mi">500</span>
<span class="n">state_seq</span><span class="p">,</span> <span class="n">_</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">seq_len</span><span class="p">)</span>

<span class="n">counts</span> <span class="o">=</span> <span class="n">compute_counts</span><span class="p">(</span><span class="n">state_seq</span><span class="p">,</span> <span class="n">nstates</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">counts</span><span class="p">)</span>

<span class="n">trans_mat_empirical</span> <span class="o">=</span> <span class="n">normalize_counts</span><span class="p">(</span><span class="n">counts</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">trans_mat_empirical</span><span class="p">)</span>

<span class="k">assert</span> <span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">trans_mat</span><span class="p">,</span> <span class="n">trans_mat_empirical</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="mf">1e-1</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">NameError</span><span class="g g-Whitespace">                                 </span>Traceback (most recent call last)
<span class="o">&lt;</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">6</span><span class="o">-</span><span class="n">f054683fcd82</span><span class="o">&gt;</span> <span class="ow">in</span> <span class="o">&lt;</span><span class="n">module</span><span class="o">&gt;</span>
<span class="g g-Whitespace">     </span><span class="mi">30</span> <span class="n">rng_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">0</span><span class="p">)</span>
<span class="g g-Whitespace">     </span><span class="mi">31</span> <span class="n">seq_len</span> <span class="o">=</span> <span class="mi">500</span>
<span class="ne">---&gt; </span><span class="mi">32</span> <span class="n">state_seq</span><span class="p">,</span> <span class="n">_</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">seq_len</span><span class="p">)</span>
<span class="g g-Whitespace">     </span><span class="mi">33</span> 
<span class="g g-Whitespace">     </span><span class="mi">34</span> <span class="n">counts</span> <span class="o">=</span> <span class="n">compute_counts</span><span class="p">(</span><span class="n">state_seq</span><span class="p">,</span> <span class="n">nstates</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>

<span class="ne">NameError</span>: name &#39;PRNGKey&#39; is not defined
</pre></div>
</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 class="section" id="example-lillypad-hmm">
<span id="sec-lillypad"></span><h2>Example: Lillypad HMM<a class="headerlink" href="#example-lillypad-hmm" title="Permalink to this headline">¶</a></h2>
<p>If <span class="math notranslate nohighlight">\(\obs_t\)</span> is continuous, it is common to use a Gaussian
observation model:</p>
<div class="math notranslate nohighlight">
\[p(\obs_t|\hidden_t=j) = \gauss(\obs_t|\vmu_j,\vSigma_j)\]</div>
<p>This is sometimes called a Gaussian HMM.</p>
<p>As a simple example, suppose we have an HMM with 3 hidden states,
each of which generates a 2d Gaussian.
We can represent these Gaussian distributions are 2d ellipses,
as we show below.
We call these ``lilly pads’’, because of their shape.
We can imagine a frog hopping from one lilly pad to another.
(This analogy is due to the late Sam Roweis.)
The frog will stay on a pad for a while (corresponding to remaining in the same
discrete state <span class="math notranslate nohighlight">\(\hidden_t\)</span>), and then jump to a new pad
(corresponding to a transition to a new state).
The data we see are just the 2d points (e.g., water droplets)
coming from near the pad that the frog is currently on.
Thus this model is like a Gaussian mixture model,
in that it generates clusters of observations,
except now there is temporal correlation between the data points.</p>
<p>Let us now illustrate 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"># Let us create the model</span>

<span class="n">initial_probs</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="mf">0.3</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">])</span>

<span class="c1"># transition matrix</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="mf">0.3</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">]</span>
<span class="p">])</span>

<span class="c1"># Observation model</span>
<span class="n">mu_collection</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="mf">0.3</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">]</span>
<span class="p">])</span>

<span class="n">S1</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="mf">1.1</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="mf">0.3</span><span class="p">]])</span>
<span class="n">S2</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="mf">0.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.3</span><span class="p">]])</span>
<span class="n">S3</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="mf">0.8</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">]])</span>
<span class="n">cov_collection</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="n">S1</span><span class="p">,</span> <span class="n">S2</span><span class="p">,</span> <span class="n">S3</span><span class="p">])</span> <span class="o">/</span> <span class="mi">60</span>


<span class="kn">import</span> <span class="nn">tensorflow_probability</span> <span class="k">as</span> <span class="nn">tfp</span>

<span class="k">if</span> <span class="kc">False</span><span class="p">:</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">initial_probs</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">MultivariateNormalFullCovariance</span><span class="p">(</span>
                <span class="n">loc</span><span class="o">=</span><span class="n">mu_collection</span><span class="p">,</span> <span class="n">covariance_matrix</span><span class="o">=</span><span class="n">cov_collection</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</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">initial_probs</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">as_distribution</span><span class="p">(</span>
                  <span class="n">tfp</span><span class="o">.</span><span class="n">substrates</span><span class="o">.</span><span class="n">jax</span><span class="o">.</span><span class="n">distributions</span><span class="o">.</span><span class="n">MultivariateNormalFullCovariance</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="n">mu_collection</span><span class="p">,</span>
                                                                                    <span class="n">covariance_matrix</span><span class="o">=</span><span class="n">cov_collection</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 stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;distrax._src.utils.hmm.HMM object at 0x7fd658f7b7f0&gt;
</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="n">n_samples</span><span class="p">,</span> <span class="n">seed</span> <span class="o">=</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">10</span>
<span class="n">samples_state</span><span class="p">,</span> <span class="n">samples_obs</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="nb">print</span><span class="p">(</span><span class="n">samples_state</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">samples_obs</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(50,)
(50, 2)
</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"># Let&#39;s plot the observed data in 2d</span>
<span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span>
<span class="n">ymin</span><span class="p">,</span> <span class="n">ymax</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">1.2</span>
<span class="n">colors</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;tab:green&quot;</span><span class="p">,</span> <span class="s2">&quot;tab:blue&quot;</span><span class="p">,</span> <span class="s2">&quot;tab:red&quot;</span><span class="p">]</span>

<span class="k">def</span> <span class="nf">plot_2dhmm</span><span class="p">(</span><span class="n">hmm</span><span class="p">,</span> <span class="n">samples_obs</span><span class="p">,</span> <span class="n">samples_state</span><span class="p">,</span> <span class="n">colors</span><span class="p">,</span> <span class="n">ax</span><span class="p">,</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="n">ymin</span><span class="p">,</span> <span class="n">ymax</span><span class="p">,</span> <span class="n">step</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">):</span>
    <span class="n">obs_dist</span> <span class="o">=</span> <span class="n">hmm</span><span class="o">.</span><span class="n">obs_dist</span>
    <span class="n">color_sample</span> <span class="o">=</span> <span class="p">[</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">samples_state</span><span class="p">]</span>

    <span class="n">xs</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="n">step</span><span class="p">)</span>
    <span class="n">ys</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">ymin</span><span class="p">,</span> <span class="n">ymax</span><span class="p">,</span> <span class="n">step</span><span class="p">)</span>

    <span class="n">v_prob</span> <span class="o">=</span> <span class="n">vmap</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">obs_dist</span><span class="o">.</span><span class="n">prob</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])),</span> <span class="n">in_axes</span><span class="o">=</span><span class="p">(</span><span class="kc">None</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
    <span class="n">z</span> <span class="o">=</span> <span class="n">vmap</span><span class="p">(</span><span class="n">v_prob</span><span class="p">,</span> <span class="n">in_axes</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="kc">None</span><span class="p">))(</span><span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">)</span>

    <span class="n">grid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mgrid</span><span class="p">[</span><span class="n">xmin</span><span class="p">:</span><span class="n">xmax</span><span class="p">:</span><span class="n">step</span><span class="p">,</span> <span class="n">ymin</span><span class="p">:</span><span class="n">ymax</span><span class="p">:</span><span class="n">step</span><span class="p">]</span>

    <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">colors</span><span class="p">):</span>
        <span class="n">ax</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="o">*</span><span class="n">grid</span><span class="p">,</span> <span class="n">z</span><span class="p">[:,</span> <span class="p">:,</span> <span class="n">k</span><span class="p">],</span> <span class="n">levels</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">colors</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">linewidths</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
        <span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">obs_dist</span><span class="o">.</span><span class="n">mean</span><span class="p">()[</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="mf">0.13</span><span class="p">),</span> <span class="sa">f</span><span class="s2">&quot;$k$=</span><span class="si">{</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">13</span><span class="p">,</span> <span class="n">horizontalalignment</span><span class="o">=</span><span class="s2">&quot;right&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">samples_obs</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="o">*</span><span class="n">samples_obs</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">color_sample</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.8</span><span class="p">)</span>

    <span class="k">return</span> <span class="n">ax</span><span class="p">,</span> <span class="n">color_sample</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">_</span><span class="p">,</span> <span class="n">color_sample</span> <span class="o">=</span> <span class="n">plot_2dhmm</span><span class="p">(</span><span class="n">hmm</span><span class="p">,</span> <span class="n">samples_obs</span><span class="p">,</span> <span class="n">samples_state</span><span class="p">,</span> <span class="n">colors</span><span class="p">,</span> <span class="n">ax</span><span class="p">,</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="n">ymin</span><span class="p">,</span> <span class="n">ymax</span><span class="p">)</span>
</pre></div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Let&#39;s plot the hidden state sequence</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">ax</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">),</span> <span class="n">samples_state</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="n">c</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">),</span> <span class="n">samples_state</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">color_sample</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;matplotlib.collections.PathCollection at 0x7fd5e174b460&gt;
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