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   State Space Models: A Modern Approach
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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>

<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="linear-gaussian-ssms">
<span id="sec-lds-intro"></span><h1>Linear Gaussian SSMs<a class="headerlink" href="#linear-gaussian-ssms" title="Permalink to this headline">¶</a></h1>
<p>Consider the state space model in
<a class="reference internal" href="ssm_intro.html#equation-eq-ssm-ar">(1)</a>
where we assume the observations are conditionally iid given the
hidden states and inputs (i.e. there are no auto-regressive dependencies
between the observables).
We can rewrite this model as
a stochastic <span class="math notranslate nohighlight">\(\keyword{nonlinear dynamical system}\)</span> or <span class="math notranslate nohighlight">\(\keyword{NLDS}\)</span>
by defining the distribution of the next hidden state
<span class="math notranslate nohighlight">\(\hidden_t \in \real^{\nhidden}\)</span>
as a deterministic function of the past state
<span class="math notranslate nohighlight">\(\hidden_{t-1}\)</span>,
the input <span class="math notranslate nohighlight">\(\inputs_t \in \real^{\ninputs}\)</span>,
and some random <span class="math notranslate nohighlight">\(\keyword{process noise}\)</span> <span class="math notranslate nohighlight">\(\transNoise_t \in \real^{\nhidden}\)</span></p>
<div class="amsmath math notranslate nohighlight" id="equation-b48061b1-1d75-4952-be1e-9145adb38f90">
<span class="eqno">(4)<a class="headerlink" href="#equation-b48061b1-1d75-4952-be1e-9145adb38f90" title="Permalink to this equation">¶</a></span>\[\begin{align}
\hidden_t &amp;= \dynamicsFn(\hidden_{t-1}, \inputs_t, \transNoise_t)  
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(\transNoise_t\)</span> is drawn from the distribution such
that the induced distribution
on <span class="math notranslate nohighlight">\(\hidden_t\)</span> matches <span class="math notranslate nohighlight">\(p(\hidden_t|\hidden_{t-1}, \inputs_t)\)</span>.
Similarly we can rewrite the observation distribution
as a deterministic function of the hidden state
plus <span class="math notranslate nohighlight">\(\keyword{observation noise}\)</span> <span class="math notranslate nohighlight">\(\obsNoise_t \in \real^{\nobs}\)</span>:</p>
<div class="amsmath math notranslate nohighlight" id="equation-8503a723-5c54-4693-835d-f2ba58d0e341">
<span class="eqno">(5)<a class="headerlink" href="#equation-8503a723-5c54-4693-835d-f2ba58d0e341" title="Permalink to this equation">¶</a></span>\[\begin{align}
\obs_t &amp;= \measurementFn(\hidden_{t}, \inputs_t, \obsNoise_t)
\end{align}\]</div>
<p>If we assume additive Gaussian noise,
the model becomes</p>
<div class="amsmath math notranslate nohighlight" id="equation-05b276c6-1300-4f5e-832c-6ef78b230ee5">
<span class="eqno">(6)<a class="headerlink" href="#equation-05b276c6-1300-4f5e-832c-6ef78b230ee5" title="Permalink to this equation">¶</a></span>\[\begin{align}
\hidden_t &amp;= \dynamicsFn(\hidden_{t-1}, \inputs_t) +  \transNoise_t  \\
\obs_t &amp;= \measurementFn(\hidden_{t}, \inputs_t) + \obsNoise_t
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(\transNoise_t \sim \gauss(\vzero,\transCov_t)\)</span>
and <span class="math notranslate nohighlight">\(\obsNoise_t \sim \gauss(\vzero,\obsCov_t)\)</span>.
We will call these <span class="math notranslate nohighlight">\(\keyword{Gaussian SSMs}\)</span>.</p>
<p>If we additionally assume
the transition function <span class="math notranslate nohighlight">\(\dynamicsFn\)</span>
and the observation function <span class="math notranslate nohighlight">\(\measurementFn\)</span> are both linear,
then we can rewrite the model as follows:</p>
<div class="amsmath math notranslate nohighlight" id="equation-48743a9b-4689-4d37-9b6e-639de4080d17">
<span class="eqno">(7)<a class="headerlink" href="#equation-48743a9b-4689-4d37-9b6e-639de4080d17" title="Permalink to this equation">¶</a></span>\[\begin{align}
p(\hidden_t|\hidden_{t-1},\inputs_t) &amp;= \gauss(\hidden_t|\ldsDyn \hidden_{t-1}
+ \ldsDynIn \inputs_t, \transCov)
\\
p(\obs_t|\hidden_t,\inputs_t) &amp;= \gauss(\obs_t|\ldsObs \hidden_{t}
+ \ldsObsIn \inputs_t, \obsCov)
\end{align}\]</div>
<p>This is called a
<span class="math notranslate nohighlight">\(\keyword{linear-Gaussian state space model}\)</span>
or <span class="math notranslate nohighlight">\(\keyword{LG-SSM}\)</span>;
it is also called
a <span class="math notranslate nohighlight">\(\keyword{linear dynamical system}\)</span> or <span class="math notranslate nohighlight">\(\keyword{LDS}\)</span>.
We usually assume the parameters are independent of time, in which case
the model is said to be time-invariant or homogeneous.</p>
<div class="section" id="example-tracking-a-2d-point">
<span id="sec-kalman-tracking"></span><span id="sec-tracking-lds"></span><h2>Example: tracking a 2d point<a class="headerlink" href="#example-tracking-a-2d-point" title="Permalink to this headline">¶</a></h2>
<p>Consider an object moving in <span class="math notranslate nohighlight">\(\real^2\)</span>.
Let the state be
the position and velocity of the object,
<span class="math notranslate nohighlight">\(\hidden_t =\begin{pmatrix} u_t &amp; \dot{u}_t &amp; v_t &amp; \dot{v}_t \end{pmatrix}\)</span>.
(We use <span class="math notranslate nohighlight">\(u\)</span> and <span class="math notranslate nohighlight">\(v\)</span> for the two coordinates,
to avoid confusion with the state and observation variables.)
If we use Euler discretization,
the dynamics become</p>
<div class="amsmath math notranslate nohighlight" id="equation-91ff5c5d-8e5f-4499-aefa-5879e12171e0">
<span class="eqno">(8)<a class="headerlink" href="#equation-91ff5c5d-8e5f-4499-aefa-5879e12171e0" title="Permalink to this equation">¶</a></span>\[\begin{align}
\underbrace{\begin{pmatrix} u_t\\ \dot{u}_t \\ v_t \\ \dot{v}_t \end{pmatrix}}_{\hidden_t}
  = 
\underbrace{
\begin{pmatrix}
1 &amp; 0 &amp; \Delta &amp; 0 \\
0 &amp; 1 &amp; 0 &amp; \Delta\\
0 &amp; 0 &amp; 1 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 1
\end{pmatrix}
}_{\ldsDyn}

\underbrace{\begin{pmatrix} u_{t-1} \\ \dot{u}_{t-1} \\ v_{t-1} \\ \dot{v}_{t-1} \end{pmatrix}}_{\hidden_{t-1}}
+ \transNoise_t
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(\transNoise_t \sim \gauss(\vzero,\transCov)\)</span> is
the process noise.
We assume
that the process noise is
a white noise process added to the velocity components
of the state, but not to the location,
so <span class="math notranslate nohighlight">\(\transCov = \diag(0, q, 0, q)\)</span>.
This is known as a random accelerations model.
(See  <span id="id1">[<a class="reference internal" href="../../bib.html#id18" title="Simo Sarkka. Bayesian Filtering and Smoothing. Cambridge University Press, 2013. URL: https://users.aalto.fi/~ssarkka/pub/cup_book_online_20131111.pdf.">Sar13</a>]</span> p60 for a more accurate way
to convert the continuous time process to discrete time.)</p>
<p>Now suppose that at each discrete time point we
observe the location,
corrupted by  Gaussian noise.
Thus the observation model becomes</p>
<div class="amsmath math notranslate nohighlight" id="equation-57a3bfcc-a07e-4569-b6ea-c293d7f66be7">
<span class="eqno">(9)<a class="headerlink" href="#equation-57a3bfcc-a07e-4569-b6ea-c293d7f66be7" title="Permalink to this equation">¶</a></span>\[\begin{align}
\underbrace{\begin{pmatrix}  \obs_{1,t} \\  \obs_{2,t} \end{pmatrix}}_{\obs_t}
  &amp;=
    \underbrace{
    \begin{pmatrix}
1 &amp; 0 &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; 1 &amp; 0
    \end{pmatrix}
    }_{\ldsObs}
    
\underbrace{\begin{pmatrix} u_t\\ \dot{u}_t \\ v_t \\ \dot{v}_t \end{pmatrix}}_{\hidden_t}    
 + \obsNoise_t
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(\obsNoise_t \sim \gauss(\vzero,\obsCov)\)</span> is the observation noise.
We see that the observation matrix <span class="math notranslate nohighlight">\(\ldsObs\)</span> simply ``extracts’’ the
relevant parts  of the state vector.</p>
<p>Suppose we sample a trajectory and corresponding set
of noisy observations from this model,
<span class="math notranslate nohighlight">\((\hidden_{1:T}, \obs_{1:T}) \sim p(\hidden,\obs|\params)\)</span>.
(We use diagonal observation noise,
<span class="math notranslate nohighlight">\(\obsCov = \diag(\sigma_1^2, \sigma_2^2)\)</span>.)
The results are shown below.</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">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">314</span><span class="p">)</span>
<span class="n">timesteps</span> <span class="o">=</span> <span class="mi">15</span>
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span>
    <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">delta</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="p">])</span>

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

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

<span class="n">Q</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">state_size</span><span class="p">)</span> <span class="o">*</span> <span class="mf">0.001</span>
<span class="n">R</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">observation_size</span><span class="p">)</span> <span class="o">*</span> <span class="mf">1.0</span>
<span class="c1"># Prior parameter distribution</span>
<span class="n">mu0</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">8</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
<span class="n">Sigma0</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">state_size</span><span class="p">)</span> <span class="o">*</span> <span class="mf">1.0</span>

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

<span class="n">lds</span> <span class="o">=</span> <span class="n">LDS</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">mu0</span><span class="p">,</span> <span class="n">Sigma0</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">lds</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>LDS(A=DeviceArray([[1., 0., 1., 0.],
             [0., 1., 0., 1.],
             [0., 0., 1., 0.],
             [0., 0., 0., 1.]], dtype=float32), C=DeviceArray([[1, 0, 0, 0],
             [0, 1, 0, 0]], dtype=int32), Q=DeviceArray([[0.001, 0.   , 0.   , 0.   ],
             [0.   , 0.001, 0.   , 0.   ],
             [0.   , 0.   , 0.001, 0.   ],
             [0.   , 0.   , 0.   , 0.001]], dtype=float32), R=DeviceArray([[1., 0.],
             [0., 1.]], dtype=float32), mu=DeviceArray([ 8., 10.,  1.,  0.], dtype=float32), Sigma=DeviceArray([[1., 0., 0., 0.],
             [0., 1., 0., 0.],
             [0., 0., 1., 0.],
             [0., 0., 0., 1.]], dtype=float32), state_offset=None, obs_offset=None, nstates=4, nobs=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="kn">from</span> <span class="nn">jsl.demos.plot_utils</span> <span class="kn">import</span> <span class="n">plot_ellipse</span>

<span class="k">def</span> <span class="nf">plot_tracking_values</span><span class="p">(</span><span class="n">observed</span><span class="p">,</span> <span class="n">filtered</span><span class="p">,</span> <span class="n">cov_hist</span><span class="p">,</span> <span class="n">signal_label</span><span class="p">,</span> <span class="n">ax</span><span class="p">):</span>
    <span class="n">timesteps</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">observed</span><span class="o">.</span><span class="n">shape</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">observed</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">observed</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
            <span class="n">markerfacecolor</span><span class="o">=</span><span class="s2">&quot;none&quot;</span><span class="p">,</span> <span class="n">markeredgewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;observed&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="o">*</span><span class="n">filtered</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">signal_label</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;tab:red&quot;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;x&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
    <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">timesteps</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
        <span class="n">covn</span> <span class="o">=</span> <span class="n">cov_hist</span><span class="p">[</span><span class="n">t</span><span class="p">][:</span><span class="mi">2</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span>
        <span class="n">plot_ellipse</span><span class="p">(</span><span class="n">covn</span><span class="p">,</span> <span class="n">filtered</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">],</span> <span class="n">ax</span><span class="p">,</span> <span class="n">n_std</span><span class="o">=</span><span class="mf">2.0</span><span class="p">,</span> <span class="n">plot_center</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">&quot;equal&quot;</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
</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">z_hist</span><span class="p">,</span> <span class="n">x_hist</span> <span class="o">=</span> <span class="n">lds</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">timesteps</span><span class="p">)</span>

<span class="n">fig_truth</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">axs</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_hist</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">x_hist</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
        <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">markerfacecolor</span><span class="o">=</span><span class="s2">&quot;none&quot;</span><span class="p">,</span>
        <span class="n">markeredgewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span>
        <span class="n">label</span><span class="o">=</span><span class="s2">&quot;observed&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">)</span>

<span class="n">axs</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z_hist</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">z_hist</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
        <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;truth&quot;</span><span class="p">,</span>
        <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;s&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">8</span><span class="p">)</span>
<span class="n">axs</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">axs</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">&quot;equal&quot;</span><span class="p">)</span>
</pre></div>
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<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(7.24486608505249, 23.857812213897706, 8.042076778411865, 11.636079120635987)
</pre></div>
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<img alt="../../_images/lds_5_1.png" src="../../_images/lds_5_1.png" />
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<p>The main task is to infer the hidden states given the noisy
observations, i.e., <span class="math notranslate nohighlight">\(p(\hidden_t|\obs_{1:t},\params)\)</span>
or <span class="math notranslate nohighlight">\(p(\hidden_t|\obs_{1:T}, \params)\)</span> in the offline case.
We discuss the topic of inference in <a class="reference internal" href="inference.html#sec-inference"><span class="std std-ref">States estimation (inference)</span></a>.
We will usually represent this belief state by a Gaussian distribution,
<span class="math notranslate nohighlight">\(p(\hidden_t|\obs_{1:s},\params) = \gauss(\hidden_t| \mean_{t|s}, \covMat_{t|s})\)</span>,
where usually <span class="math notranslate nohighlight">\(s=t\)</span> or <span class="math notranslate nohighlight">\(s=T\)</span>.
Sometimes we use information form,
<span class="math notranslate nohighlight">\(p(\hidden_t|\obs_{1:s},\params) = \gaussInfo(\hidden_t|\precMean_{t|s}, \precMat_{t|s})\)</span>.</p>
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