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   State Space Models: A Modern Approach
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   State Space Models
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     What are State Space Models?
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     Hidden Markov Models
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     Linear Gaussian SSMs
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     Kalman filtering
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     Kalman (RTS) smoother
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     Extended Kalman smoother
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     Parallel extended Kalman smoothing
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   Sequential Monte Carlo
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   Offline parameter estimation (learning)
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   Data assimilation using Ensemble Kalman filter
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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="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">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="cell docutils container">
<div class="cell_input docutils container">
<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="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">ssm_jax</span>
<span class="kn">from</span> <span class="nn">ssm_jax.hmm.models</span> <span class="kn">import</span> <span class="n">GaussianHMM</span>

<span class="n">print_source</span><span class="p">(</span><span class="n">GaussianHMM</span><span class="p">)</span>
</pre></div>
</div>
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<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">class GaussianHMM<span style="font-weight: bold">(</span>BaseHMM<span style="font-weight: bold">)</span>:
    def __init__<span style="font-weight: bold">(</span>self,
                 initial_probabilities,
                 transition_matrix,
                 emission_means,
                 emission_covariance_matrices<span style="font-weight: bold">)</span>:
        <span style="color: #008000; text-decoration-color: #008000">""</span>"_summary_

        Args:
            initial_probabilities <span style="font-weight: bold">(</span>_type_<span style="font-weight: bold">)</span>: _description_
            transition_matrix <span style="font-weight: bold">(</span>_type_<span style="font-weight: bold">)</span>: _description_
            emission_means <span style="font-weight: bold">(</span>_type_<span style="font-weight: bold">)</span>: _description_
            emission_covariance_matrices <span style="font-weight: bold">(</span>_type_<span style="font-weight: bold">)</span>: _description_
        <span style="color: #008000; text-decoration-color: #008000">""</span>"
        super<span style="font-weight: bold">()</span>.__init__<span style="font-weight: bold">(</span>initial_probabilities,
                         transition_matrix<span style="font-weight: bold">)</span>

        self._emission_distribution = tfd.MultivariateNormalFullCovariance<span style="font-weight: bold">(</span>
            emission_means, emission_covariance_matrices<span style="font-weight: bold">)</span>

    @classmethod
    def random_initialization<span style="font-weight: bold">(</span>cls, key, num_states, emission_dim<span style="font-weight: bold">)</span>:
        key1, key2, key3 = jr.split<span style="font-weight: bold">(</span>key, <span style="color: #000080; text-decoration-color: #000080; font-weight: bold">3</span><span style="font-weight: bold">)</span>
        initial_probs = jr.dirichlet<span style="font-weight: bold">(</span>key1, jnp.ones<span style="font-weight: bold">(</span>num_states<span style="font-weight: bold">))</span>
        transition_matrix = jr.dirichlet<span style="font-weight: bold">(</span>key2, jnp.ones<span style="font-weight: bold">(</span>num_states<span style="font-weight: bold">)</span>, <span style="font-weight: bold">(</span>num_states,<span style="font-weight: bold">))</span>
        emission_means = jr.normal<span style="font-weight: bold">(</span>key3, <span style="font-weight: bold">(</span>num_states, emission_dim<span style="font-weight: bold">))</span>
        emission_covs = jnp.tile<span style="font-weight: bold">(</span>jnp.eye<span style="font-weight: bold">(</span>emission_dim<span style="font-weight: bold">)</span>, <span style="font-weight: bold">(</span>num_states, <span style="color: #000080; text-decoration-color: #000080; font-weight: bold">1</span>, <span style="color: #000080; text-decoration-color: #000080; font-weight: bold">1</span><span style="font-weight: bold">))</span>
        return cls<span style="font-weight: bold">(</span>initial_probs, transition_matrix, emission_means, emission_covs<span style="font-weight: bold">)</span>

    # Properties to get various parameters of the model
    @property
    def emission_distribution<span style="font-weight: bold">(</span>self<span style="font-weight: bold">)</span>:
        return self._emission_distribution

    @property
    def emission_means<span style="font-weight: bold">(</span>self<span style="font-weight: bold">)</span>:
        return self.emission_distribution.mean<span style="font-weight: bold">()</span>

    @property
    def emission_covariance_matrices<span style="font-weight: bold">(</span>self<span style="font-weight: bold">)</span>:
        return self.emission_distribution.covariance<span style="font-weight: bold">()</span>

    @property
    def unconstrained_params<span style="font-weight: bold">(</span>self<span style="font-weight: bold">)</span>:
        <span style="color: #008000; text-decoration-color: #008000">""</span>"Helper property to get a PyTree of unconstrained parameters.
        <span style="color: #008000; text-decoration-color: #008000">""</span>"
        return tfb.SoftmaxCentered<span style="font-weight: bold">()</span>.inverse<span style="font-weight: bold">(</span>self.initial_probabilities<span style="font-weight: bold">)</span>, \
               tfb.SoftmaxCentered<span style="font-weight: bold">()</span>.inverse<span style="font-weight: bold">(</span>self.transition_matrix<span style="font-weight: bold">)</span>, \
               self.emission_means, \
               PSDToRealBijector.forward<span style="font-weight: bold">(</span>self.emission_covariance_matrices<span style="font-weight: bold">)</span>

    @classmethod
    def from_unconstrained_params<span style="font-weight: bold">(</span>cls, unconstrained_params, hypers<span style="font-weight: bold">)</span>:
        initial_probabilities = tfb.SoftmaxCentered<span style="font-weight: bold">()</span>.forward<span style="font-weight: bold">(</span>unconstrained_params<span style="font-weight: bold">[</span><span style="color: #000080; text-decoration-color: #000080; font-weight: bold">0</span><span style="font-weight: bold">])</span>
        transition_matrix = tfb.SoftmaxCentered<span style="font-weight: bold">()</span>.forward<span style="font-weight: bold">(</span>unconstrained_params<span style="font-weight: bold">[</span><span style="color: #000080; text-decoration-color: #000080; font-weight: bold">1</span><span style="font-weight: bold">])</span>
        emission_means = unconstrained_params<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>
        emission_covs = PSDToRealBijector.inverse<span style="font-weight: bold">(</span>unconstrained_params<span style="font-weight: bold">[</span><span style="color: #000080; text-decoration-color: #000080; font-weight: bold">3</span><span style="font-weight: bold">])</span>
        return cls<span style="font-weight: bold">(</span>initial_probabilities, transition_matrix, emission_means, emission_covs, 
*hypers<span style="font-weight: bold">)</span>

</pre>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set dimensions</span>
<span class="n">num_states</span> <span class="o">=</span> <span class="mi">5</span>
<span class="n">emission_dim</span> <span class="o">=</span> <span class="mi">2</span>

<span class="c1"># Specify parameters of the HMM</span>
<span class="n">initial_probs</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">num_states</span><span class="p">)</span> <span class="o">/</span> <span class="n">num_states</span>
<span class="n">transition_matrix</span> <span class="o">=</span> <span class="mf">0.95</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">num_states</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.05</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">num_states</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">emission_means</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">column_stack</span><span class="p">([</span>
    <span class="n">jnp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">num_states</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="p">],</span>
    <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">num_states</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="p">]</span>
<span class="p">])</span>
<span class="n">emission_covs</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span><span class="mf">0.1</span><span class="o">**</span><span class="mi">2</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">emission_dim</span><span class="p">),</span> <span class="p">(</span><span class="n">num_states</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>

<span class="n">hmm</span> <span class="o">=</span> <span class="n">GaussianHMM</span><span class="p">(</span><span class="n">initial_probs</span><span class="p">,</span>
                       <span class="n">transition_matrix</span><span class="p">,</span>
                       <span class="n">emission_means</span><span class="p">,</span> 
                       <span class="n">emission_covs</span><span class="p">)</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>
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<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 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, key, num_timesteps<span style="font-weight: bold">)</span>:
        <span style="color: #008000; text-decoration-color: #008000">""</span>"Sample a sequence of latent states and emissions.

        Args:
            key <span style="font-weight: bold">(</span>_type_<span style="font-weight: bold">)</span>: _description_
            num_timesteps <span style="font-weight: bold">(</span>_type_<span style="font-weight: bold">)</span>: _description_
        <span style="color: #008000; text-decoration-color: #008000">""</span>"
        def _step<span style="font-weight: bold">(</span>state, key<span style="font-weight: bold">)</span>:
            key1, key2 = jr.split<span style="font-weight: bold">(</span>key, <span style="color: #000080; text-decoration-color: #000080; font-weight: bold">2</span><span style="font-weight: bold">)</span>
            emission = self.emission_distribution.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">key1</span><span style="font-weight: bold">)</span>
            next_state = self.transition_distribution.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">key2</span><span style="font-weight: bold">)</span>
            return next_state, <span style="font-weight: bold">(</span>state, emission<span style="font-weight: bold">)</span>

        # Sample the initial state
        key1, key = jr.split<span style="font-weight: bold">(</span>key, <span style="color: #000080; text-decoration-color: #000080; font-weight: bold">2</span><span style="font-weight: bold">)</span>
        initial_state = self.initial_distribution.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">key1</span><span style="font-weight: bold">)</span>

        # Sample the remaining emissions and states
        keys = jr.split<span style="font-weight: bold">(</span>key, num_timesteps<span style="font-weight: bold">)</span>
        _, <span style="font-weight: bold">(</span>states, emissions<span style="font-weight: bold">)</span> = lax.scan<span style="font-weight: bold">(</span>_step, initial_state, keys<span style="font-weight: bold">)</span>
        return states, emissions

</pre>
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<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">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="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>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;distrax._src.utils.hmm.HMM object at 0x7fde82c856d0&gt;
</pre></div>
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<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>
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<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>
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