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   State Space Models: A Modern Approach
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\newcommand{\vwh}{\hat{\vw}}
\newcommand{\vx}{\mathbf{x}}
%\newcommand{\vx}{\mathbf{x}}
\newcommand{\vxt}{\mathbf{\tilde{x}}}
\newcommand{\vy}{\mathbf{y}}
\newcommand{\vyt}{\mathbf{\tilde{y}}}
\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{\hmmhid}{\hidden}
\newcommand{\hmmobs}{\obs}\\\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="tex2jax_ignore mathjax_ignore section" id="what-are-state-space-models">
<span id="sec-ssm-intro"></span><h1>What are State Space Models?<a class="headerlink" href="#what-are-state-space-models" title="Permalink to this headline">¶</a></h1>
<p>A state space model or SSM
is a partially observed Markov model,
in which the hidden state,  <span class="math notranslate nohighlight">\(\hidden_t\)</span>,
evolves over time according to a Markov process,
possibly conditional on external inputs or controls <span class="math notranslate nohighlight">\(\input_t\)</span>,
and each hidden state generates some
observations <span class="math notranslate nohighlight">\(\obs_t\)</span> at each time step.
(In this book, we mostly focus on discrete time systems,
although  we consider the continuous-time case in  XXX.)
We get to see the observations, but not the hidden state.
Our main goal is to infer the hidden state given the observations.
However, we can also use the model to predict future observations,
by first predicting future hidden states, and then predicting
what observations they might generate.
By using  a hidden state <span class="math notranslate nohighlight">\(\hidden_t\)</span>
to represent the past observations, <span class="math notranslate nohighlight">\(\obs_{1:t-1}\)</span>,
the  model can have ``infinite’’ memory,
unlike a standard Markov model.</p>
<div class="figure align-default" id="fig-ssm-ar">
<a class="reference internal image-reference" href="../../_images/SSM-AR-inputs.png"><img alt="../../_images/SSM-AR-inputs.png" src="../../_images/SSM-AR-inputs.png" style="height: 300px;" /></a>
<p class="caption"><span class="caption-number">Fig. 3 </span><span class="caption-text">Illustration of an SSM as a graphical model.</span><a class="headerlink" href="#fig-ssm-ar" title="Permalink to this image">¶</a></p>
</div>
<p>Formally we can define an SSM
as the following joint distribution:</p>
<div class="math notranslate nohighlight" id="equation-eq-ssm-ar">
<span class="eqno">(2)<a class="headerlink" href="#equation-eq-ssm-ar" title="Permalink to this equation">¶</a></span>\[p(\hmmobs_{1:T},\hmmhid_{1:T}|\inputs_{1:T})
 = \left[ p(\hmmhid_1|\inputs_1) \prod_{t=2}^{T}
 p(\hmmhid_t|\hmmhid_{t-1},\inputs_t) \right]
 \left[ \prod_{t=1}^T p(\hmmobs_t|\hmmhid_t, \inputs_t, \hmmobs_{t-1}) \right]\]</div>
<p>where <span class="math notranslate nohighlight">\(p(\hmmhid_t|\hmmhid_{t-1},\inputs_t)\)</span> is the
transition model,
<span class="math notranslate nohighlight">\(p(\hmmobs_t|\hmmhid_t, \inputs_t, \hmmobs_{t-1})\)</span> is the
observation model,
and <span class="math notranslate nohighlight">\(\inputs_{t}\)</span> is an optional input or action.
See <a class="reference internal" href="#fig-ssm-ar"><span class="std std-numref">Fig. 3</span></a>
for an illustration of the corresponding graphical model.</p>
<p>We often consider a simpler setting in which the
observations are conditionally independent of each other
(rather than having Markovian dependencies) given the hidden state.
In this case the joint simplifies to</p>
<div class="math notranslate nohighlight" id="equation-eq-ssm-input">
<span class="eqno">(3)<a class="headerlink" href="#equation-eq-ssm-input" title="Permalink to this equation">¶</a></span>\[p(\hmmobs_{1:T},\hmmhid_{1:T}|\inputs_{1:T})
 = \left[ p(\hmmhid_1|\inputs_1) \prod_{t=2}^{T}
 p(\hmmhid_t|\hmmhid_{t-1},\inputs_t) \right]
 \left[ \prod_{t=1}^T p(\hmmobs_t|\hmmhid_t, \inputs_t) \right]\]</div>
<p>Sometimes there are no external inputs, so the model further
simplifies to the following unconditional generative model:</p>
<div class="math notranslate nohighlight" id="equation-eq-ssm-no-input">
<span class="eqno">(4)<a class="headerlink" href="#equation-eq-ssm-no-input" title="Permalink to this equation">¶</a></span>\[p(\hmmobs_{1:T},\hmmhid_{1:T})
 = \left[ p(\hmmhid_1) \prod_{t=2}^{T}
 p(\hmmhid_t|\hmmhid_{t-1}) \right]
 \left[ \prod_{t=1}^T p(\hmmobs_t|\hmmhid_t) \right]\]</div>
<p>See <a class="reference internal" href="#ssm-simplified"><span class="std std-numref">Fig. 4</span></a>
for an illustration of the corresponding graphical model.</p>
<div class="figure align-default" id="ssm-simplified">
<a class="reference internal image-reference" href="../../_images/SSM-simplified.png"><img alt="../../_images/SSM-simplified.png" src="../../_images/SSM-simplified.png" style="width: 136.0px; height: 98.0px;" /></a>
<p class="caption"><span class="caption-number">Fig. 4 </span><span class="caption-text">Illustration of a simplified SSM.</span><a class="headerlink" href="#ssm-simplified" title="Permalink to this image">¶</a></p>
</div>
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