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\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{\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="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="c1">#try:</span>
<span class="c1">#    import ssm_jax</span>
<span class="c1">##except:</span>
<span class="c1">#    %pip install git+https://github.com/probml/ssm-jax</span>
<span class="c1">#    import ssm_jax</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="tex2jax_ignore mathjax_ignore section" id="hmm-filtering-forwards-algorithm">
<span id="sec-forwards"></span><h1>HMM filtering (forwards algorithm)<a class="headerlink" href="#hmm-filtering-forwards-algorithm" title="Permalink to this headline">¶</a></h1>
<p>The  <strong>Bayes filter</strong> is an algorithm for recursively computing
the belief state
<span class="math notranslate nohighlight">\(p(\hidden_t|\obs_{1:t})\)</span> given
the prior belief from the previous step,
<span class="math notranslate nohighlight">\(p(\hidden_{t-1}|\obs_{1:t-1})\)</span>,
the new observation <span class="math notranslate nohighlight">\(\obs_t\)</span>,
and the model.
This can be done using <strong>sequential Bayesian updating</strong>.
For a dynamical model, this reduces to the
<strong>predict-update</strong> cycle described below.</p>
<p>The <strong>prediction step</strong> is just the <strong>Chapman-Kolmogorov equation</strong>:</p>
<div class="math notranslate nohighlight">
\[p(\hidden_t|\obs_{1:t-1})
= \int p(\hidden_t|\hidden_{t-1}) p(\hidden_{t-1}|\obs_{1:t-1}) d\hidden_{t-1}\]</div>
<p>The prediction step computes
the one-step-ahead predictive distribution
for the latent state, which updates
the posterior from the previous time step into the prior
for the current step.</p>
<p>The <strong>update step</strong>
is just Bayes rule:</p>
<div class="math notranslate nohighlight">
\[p(\hidden_t|\obs_{1:t}) = \frac{1}{Z_t}
p(\obs_t|\hidden_t) p(\hidden_t|\obs_{1:t-1})\]</div>
<p>where the normalization constant is</p>
<div class="math notranslate nohighlight">
\[Z_t = \int p(\obs_t|\hidden_t) p(\hidden_t|\obs_{1:t-1}) d\hidden_{t}
= p(\obs_t|\obs_{1:t-1})\]</div>
<p>When the latent states <span class="math notranslate nohighlight">\(\hidden_t\)</span> are discrete, as in HMM,
the above integrals become sums.
In particular, suppose we define
the belief state as <span class="math notranslate nohighlight">\(\alpha_t(j) \defeq p(\hidden_t=j|\obs_{1:t})\)</span>,
the local evidence as <span class="math notranslate nohighlight">\(\lambda_t(j) \defeq p(\obs_t|\hidden_t=j)\)</span>,
and the transition matrix
<span class="math notranslate nohighlight">\(A(i,j)  = p(\hidden_t=j|\hidden_{t-1}=i)\)</span>.
Then the predict step becomes</p>
<div class="math notranslate nohighlight" id="equation-eqn-predictivehmm">
<span class="eqno">(19)<a class="headerlink" href="#equation-eqn-predictivehmm" title="Permalink to this equation">¶</a></span>\[\alpha_{t|t-1}(j) \defeq p(\hidden_t=j|\obs_{1:t-1})
 = \sum_i \alpha_{t-1}(i) A(i,j)\]</div>
<p>and the update step becomes</p>
<div class="math notranslate nohighlight" id="equation-eqn-fwdseqn">
<span class="eqno">(20)<a class="headerlink" href="#equation-eqn-fwdseqn" title="Permalink to this equation">¶</a></span>\[\alpha_t(j)
= \frac{1}{Z_t} \lambda_t(j) \alpha_{t|t-1}(j)
= \frac{1}{Z_t} \lambda_t(j) \left[\sum_i \alpha_{t-1}(i) A(i,j)  \right]\]</div>
<p>where
the  normalization constant for each time step is given by</p>
<div class="math notranslate nohighlight" id="equation-eqn-hmmz">
<span class="eqno">(21)<a class="headerlink" href="#equation-eqn-hmmz" title="Permalink to this equation">¶</a></span>\[\begin{split}\begin{align}
Z_t \defeq p(\obs_t|\obs_{1:t-1})
&amp;=  \sum_{j=1}^K p(\obs_t|\hidden_t=j)  p(\hidden_t=j|\obs_{1:t-1}) \\
&amp;=  \sum_{j=1}^K \lambda_t(j) \alpha_{t|t-1}(j)
\end{align}\end{split}\]</div>
<p>Since all the quantities are finite length vectors and matrices,
we can write the update equation
in matrix-vector notation as follows:</p>
<div class="math notranslate nohighlight">
\[\valpha_t =\text{normalize}\left(
\vlambda_t \dotstar  (\vA^{\trans} \valpha_{t-1}) \right)
\label{eqn:fwdsAlgoMatrixForm}\]</div>
<p>where <span class="math notranslate nohighlight">\(\dotstar\)</span> represents
elementwise vector multiplication,
and the <span class="math notranslate nohighlight">\(\text{normalize}\)</span> function just ensures its argument sums to one.</p>
<p>In {ref}(sec:casino-inference)
we illustrate
filtering for the casino HMM,
applied to a random sequence <span class="math notranslate nohighlight">\(\obs_{1:T}\)</span> of length <span class="math notranslate nohighlight">\(T=300\)</span>.
In blue, we plot the probability that the dice is in the loaded (vs fair) state,
based on the evidence seen so far.
The gray bars indicate time intervals during which the generative
process actually switched to the loaded dice.
We see that the probability generally increases in the right places.</p>
<p>Here is a JAX implementation of the forwards 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="kn">import</span> <span class="nn">jsl.hmm.hmm_lib</span> <span class="k">as</span> <span class="nn">hmm_lib</span>
<span class="n">print_source</span><span class="p">(</span><span class="n">hmm_lib</span><span class="o">.</span><span class="n">hmm_forwards_jax</span><span class="p">)</span>
<span class="c1">#https://github.com/probml/JSL/blob/main/jsl/hmm/hmm_lib.py#L189</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">@jit
def hmm_forwards_jax<span style="font-weight: bold">(</span>params, obs_seq, <span style="color: #808000; text-decoration-color: #808000">length</span>=<span style="color: #800080; text-decoration-color: #800080; font-style: italic">None</span><span style="font-weight: bold">)</span>:
    <span style="color: #008000; text-decoration-color: #008000">''</span>'
    Calculates a belief state

    Parameters
    ----------
    params : HMMJax
        Hidden Markov Model

    obs_seq: array<span style="font-weight: bold">(</span>seq_len<span style="font-weight: bold">)</span>
        History of observable events

    Returns
    -------
    * float
        The loglikelihood giving log<span style="font-weight: bold">(</span>p<span style="font-weight: bold">(</span>x|model<span style="font-weight: bold">))</span>

    * array<span style="font-weight: bold">(</span>seq_len, n_hidden<span style="font-weight: bold">)</span> :
        All alpha values found for each sample
    <span style="color: #008000; text-decoration-color: #008000">''</span>'
    seq_len = len<span style="font-weight: bold">(</span>obs_seq<span style="font-weight: bold">)</span>

    if length is <span style="color: #800080; text-decoration-color: #800080; font-style: italic">None</span>:
        length = seq_len

    trans_mat, obs_mat, init_dist = params.trans_mat, params.obs_mat, params.init_dist

    trans_mat = jnp.array<span style="font-weight: bold">(</span>trans_mat<span style="font-weight: bold">)</span>
    obs_mat = jnp.array<span style="font-weight: bold">(</span>obs_mat<span style="font-weight: bold">)</span>
    init_dist = jnp.array<span style="font-weight: bold">(</span>init_dist<span style="font-weight: bold">)</span>

    n_states, n_obs = obs_mat.shape

    def scan_fn<span style="font-weight: bold">(</span>carry, t<span style="font-weight: bold">)</span>:
        <span style="font-weight: bold">(</span>alpha_prev, log_ll_prev<span style="font-weight: bold">)</span> = carry
        alpha_n = jnp.where<span style="font-weight: bold">(</span>t &lt; length,
                            obs_mat<span style="font-weight: bold">[</span>:, obs_seq<span style="font-weight: bold">]</span> * <span style="font-weight: bold">(</span>alpha_prev<span style="font-weight: bold">[</span>:, <span style="color: #800080; text-decoration-color: #800080; font-style: italic">None</span><span style="font-weight: bold">]</span> * 
trans_mat<span style="font-weight: bold">)</span>.sum<span style="font-weight: bold">(</span><span style="color: #808000; text-decoration-color: #808000">axis</span>=<span style="color: #000080; text-decoration-color: #000080; font-weight: bold">0</span><span style="font-weight: bold">)</span>,
                            jnp.zeros_like<span style="font-weight: bold">(</span>alpha_prev<span style="font-weight: bold">))</span>

        alpha_n, cn = normalize<span style="font-weight: bold">(</span>alpha_n<span style="font-weight: bold">)</span>
        carry = <span style="font-weight: bold">(</span>alpha_n, jnp.log<span style="font-weight: bold">(</span>cn<span style="font-weight: bold">)</span> + log_ll_prev<span style="font-weight: bold">)</span>

        return carry, alpha_n

    # initial belief state
    alpha_0, c0 = normalize<span style="font-weight: bold">(</span>init_dist * obs_mat<span style="font-weight: bold">[</span>:, obs_seq<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>

    # setup scan loop
    init_state = <span style="font-weight: bold">(</span>alpha_0, jnp.log<span style="font-weight: bold">(</span>c0<span style="font-weight: bold">))</span>
    ts = jnp.arange<span style="font-weight: bold">(</span><span style="color: #000080; text-decoration-color: #000080; font-weight: bold">1</span>, seq_len<span style="font-weight: bold">)</span>
    carry, alpha_hist = lax.scan<span style="font-weight: bold">(</span>scan_fn, init_state, ts<span style="font-weight: bold">)</span>

    # post-process
    alpha_hist = jnp.vstack<span style="font-weight: bold">()</span>
    <span style="font-weight: bold">(</span>alpha_final, log_ll<span style="font-weight: bold">)</span> = carry
    return log_ll, alpha_hist

</pre>
</div></div>
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