# What are State Space Models? — State Space Models: A Modern Approach

# What are State Space Models?

    # meta-data does not work yet in VScode
    # https://github.com/microsoft/vscode-jupyter/issues/1121

    {
        "tags": [
            "hide-cell"
        ]
    }

    ### Install necessary libraries

    try:
        import jax
    except:
        # For cuda version, see https://github.com/google/jax#installation
        %pip install --upgrade "jax[cpu]"
        import jax

    try:
        import distrax
    except:
        %pip install --upgrade  distrax
        import distrax

    try:
        import jsl
    except:
        %pip install git+https://github.com/probml/jsl
        import jsl

    try:
        import rich
    except:
        %pip install rich
        import rich

    {
        "tags": [
            "hide-cell"
        ]
    }

    ### Import standard libraries

    import abc
    from dataclasses import dataclass
    import functools
    import itertools

    from typing import Any, Callable, NamedTuple, Optional, Union, Tuple

    import matplotlib.pyplot as plt
    import numpy as np

    import jax
    import jax.numpy as jnp
    from jax import lax, vmap, jit, grad
    from jax.scipy.special import logit
    from jax.nn import softmax
    from functools import partial
    from jax.random import PRNGKey, split

    import inspect
    import inspect as py_inspect
    import rich
    from rich import inspect as r_inspect
    from rich import print as r_print

    def print_source(fname):
        r_print(py_inspect.getsource(fname))

\[ \begin{align}\begin{aligned}\newcommand\floor[1]{\lfloor#1\rfloor}\\\newcommand{\real}{\mathbb{R}}\\% Numbers \newcommand{\vzero}{\boldsymbol{0}} \newcommand{\vone}{\boldsymbol{1}}\\% Greek https://www.latex-tutorial.com/symbols/greek-alphabet/ \newcommand{\valpha}{\boldsymbol{\alpha}} \newcommand{\vbeta}{\boldsymbol{\beta}} \newcommand{\vchi}{\boldsymbol{\chi}} \newcommand{\vdelta}{\boldsymbol{\delta}} \newcommand{\vDelta}{\boldsymbol{\Delta}} \newcommand{\vepsilon}{\boldsymbol{\epsilon}} \newcommand{\vzeta}{\boldsymbol{\zeta}} \newcommand{\vXi}{\boldsymbol{\Xi}} \newcommand{\vell}{\boldsymbol{\ell}} \newcommand{\veta}{\boldsymbol{\eta}} %\newcommand{\vEta}{\boldsymbol{\Eta}} \newcommand{\vgamma}{\boldsymbol{\gamma}} \newcommand{\vGamma}{\boldsymbol{\Gamma}} \newcommand{\vmu}{\boldsymbol{\mu}} \newcommand{\vmut}{\boldsymbol{\tilde{\mu}}} \newcommand{\vnu}{\boldsymbol{\nu}} \newcommand{\vkappa}{\boldsymbol{\kappa}} \newcommand{\vlambda}{\boldsymbol{\lambda}} \newcommand{\vLambda}{\boldsymbol{\Lambda}} \newcommand{\vLambdaBar}{\overline{\vLambda}} %\newcommand{\vnu}{\boldsymbol{\nu}} \newcommand{\vomega}{\boldsymbol{\omega}} \newcommand{\vOmega}{\boldsymbol{\Omega}} \newcommand{\vphi}{\boldsymbol{\phi}} \newcommand{\vvarphi}{\boldsymbol{\varphi}} \newcommand{\vPhi}{\boldsymbol{\Phi}} \newcommand{\vpi}{\boldsymbol{\pi}} \newcommand{\vPi}{\boldsymbol{\Pi}} \newcommand{\vpsi}{\boldsymbol{\psi}} \newcommand{\vPsi}{\boldsymbol{\Psi}} \newcommand{\vrho}{\boldsymbol{\rho}} \newcommand{\vtheta}{\boldsymbol{\theta}} \newcommand{\vthetat}{\boldsymbol{\tilde{\theta}}} \newcommand{\vTheta}{\boldsymbol{\Theta}} \newcommand{\vsigma}{\boldsymbol{\sigma}} \newcommand{\vSigma}{\boldsymbol{\Sigma}} \newcommand{\vSigmat}{\boldsymbol{\tilde{\Sigma}}} \newcommand{\vsigmoid}{\vsigma} \newcommand{\vtau}{\boldsymbol{\tau}} \newcommand{\vxi}{\boldsymbol{\xi}}\\ % Lower Roman (Vectors) \newcommand{\va}{\mathbf{a}} \newcommand{\vb}{\mathbf{b}} \newcommand{\vBt}{\mathbf{\tilde{B}}} \newcommand{\vc}{\mathbf{c}} \newcommand{\vct}{\mathbf{\tilde{c}}} \newcommand{\vd}{\mathbf{d}} \newcommand{\ve}{\mathbf{e}} \newcommand{\vf}{\mathbf{f}} \newcommand{\vg}{\mathbf{g}} \newcommand{\vh}{\mathbf{h}} %\newcommand{\myvh}{\mathbf{h}} \newcommand{\vi}{\mathbf{i}} \newcommand{\vj}{\mathbf{j}} \newcommand{\vk}{\mathbf{k}} \newcommand{\vl}{\mathbf{l}} \newcommand{\vm}{\mathbf{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{\vws}{\vw_s} \newcommand{\vwt}{\mathbf{\tilde{w}}} \newcommand{\vWt}{\mathbf{\tilde{W}}} \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{\ssmDyn}{f} \newcommand{\ssmObs}{h}\end{aligned}\end{align} \]

# What are State Space Models?

A state space model or SSM is a partially observed Markov model, in which the hidden state, \(\hidden_t\), evolves over time according to a Markov process, possibly conditional on external inputs or controls \(\input_t\), and each hidden state generates some observations \(\obs_t\) 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 \(\hidden_t\) to represent the past observations, \(\obs_{1:t-1}\), the model can have ``infinite’’ memory, unlike a standard Markov model.

Formally we can define an SSM as the following joint distribution:

(2)\[p(\hmmobs_{1:T},\hmmhid_{1:T}|\inputs_{1:T}) = \left[ p(\hmmhid_1|\inputs_1) \prod_{t=2}^{T} p(\hmmhid_t|\hmmhid_{t-1},\inputs_t) \right] \left[ \prod_{t=1}^T p(\hmmobs_t|\hmmhid_t, \inputs_t, \hmmobs_{t-1}) \right]\]

where \(p(\hmmhid_t|\hmmhid_{t-1},\inputs_t)\) is the transition model, \(p(\hmmobs_t|\hmmhid_t, \inputs_t, \hmmobs_{t-1})\) is the observation model, and \(\inputs_{t}\) is an optional input or action. See Figure 3 for an illustration of the corresponding graphical model.

Fig. 3 Illustration of an SSM as a graphical model.

We often consider a simpler setting in which there are no external inputs, and the observations are conditionally independent of each other (rather than having Markovian dependencies) given the hidden state. In this case the joint simplifies to

(3)\[p(\hmmobs_{1:T},\hmmhid_{1:T}) = \left[ p(\hmmhid_1) \prod_{t=2}^{T} p(\hmmhid_t|\hmmhid_{t-1}) \right] \left[ \prod_{t=1}^T p(\hmmobs_t|\hmmhid_t \right]\]

See Figure 4 for an illustration of the corresponding graphical model. Compare (2) and (3).

Fig. 4 Illustration of a simplified SSM.

# Hidden Markov Models

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 \(\hmmhid_t \in \{1,\ldots, K\}\). The observations may be discrete, \(\hmmobs_t \in \{1,\ldots, C\}\), or continuous, \(\hmmobs_t \in \real^D\), or some combination, as we illustrate below. More details can be found in e.g., [CMR05, Fra08, Rab89]. For an interactive introduction, see https://nipunbatra.github.io/hmm/.

## Example: Casino HMM

To illustrate HMMs with categorical observation model, we consider the “Ocassionally dishonest casino” model from [DEKM98]. 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.

The transition model is denoted by

\[p(z_t=j|z_{t-1}=i) = \hmmTrans_{ij}\]

Here the \(i\)’th row of \(\vA\) corresponds to the outgoing distribution from state \(i\). 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 Figure 5

Fig. 5 Illustration of the casino HMM.

The observation model \(p(\obs_t|\hidden_t=j)\) has the form

\[p(\obs_t=k|\hidden_t=j) = \hmmObs_{jk} \]

This is represented by the histograms associated with each state in Illustration of the casino HMM..

Finally, the initial state distribution is denoted by

\[p(z_1=j) = \hmmInit_j\]

Collectively we denote all the parameters by \(\vtheta=(\hmmTrans, \hmmObs, \hmmInit)\).

Now let us implement this model in code.

    # state transition matrix
    A = np.array([
        [0.95, 0.05],
        [0.10, 0.90]
    ])

    # observation matrix
    B = np.array([
        [1/6, 1/6, 1/6, 1/6, 1/6, 1/6], # fair die
        [1/10, 1/10, 1/10, 1/10, 1/10, 5/10] # loaded die
    ])

    pi = np.array([0.5, 0.5])

    (nstates, nobs) = np.shape(B)

    import distrax
    from distrax import HMM

    hmm = HMM(trans_dist=distrax.Categorical(probs=A),
                init_dist=distrax.Categorical(probs=pi),
                obs_dist=distrax.Categorical(probs=B))

    print(hmm)

    WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

    <distrax._src.utils.hmm.HMM object at 0x7fb570b7e1f0>

Let’s sample from the model. We will generate a sequence of latent states, \(\hid_{1:T}\), which we then convert to a sequence of observations, \(\obs_{1:T}\).

    seed = 314
    n_samples = 300
    z_hist, x_hist = hmm.sample(seed=PRNGKey(seed), seq_len=n_samples)

    z_hist_str = "".join((np.array(z_hist) + 1).astype(str))[:60]
    x_hist_str = "".join((np.array(x_hist) + 1).astype(str))[:60]

    print("Printing sample observed/latent...")
    print(f"x: {x_hist_str}")
    print(f"z: {z_hist_str}")

    Printing sample observed/latent...
    x: 633665342652353616444236412331351246651613325161656366246242
    z: 222222211111111111111111111111111111111222111111112222211111

    # Here is the source code for the sampling algorithm.

    print_source(hmm.sample)

      def sample(self,
                 *,
                 seed: chex.PRNGKey,
                 seq_len: chex.Array) -> Tuple:
        """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 (2,) and dtype uint32.
          seq_len: The length of the observation sequence.

        Returns:
          Tuple of hidden state sequence, and observation sequence.
        """
        rng_key, rng_init = jax.random.split(seed)
        initial_state = self._init_dist.sample(seed=rng_init)

        def draw_state(prev_state, key):
          state = self._trans_dist.sample(seed=key)
          return state, state

        rng_state, rng_obs = jax.random.split(rng_key)
        keys = jax.random.split(rng_state, seq_len - 1)
        _, states = jax.lax.scan(draw_state, initial_state, keys)
        states = jnp.append(initial_state, states)

        def draw_obs(state, key):
          return self._obs_dist.sample(seed=key)

        keys = jax.random.split(rng_obs, seq_len)
        obs_seq = jax.vmap(draw_obs, in_axes=(0, 0))(states, keys)

        return states, obs_seq

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.

# Linear Gaussian SSMs

Blah blah

## Example: model for 2d tracking

Blah blah

# Inferential goals

Fig. 6 Illustration of the different kinds of inference in an SSM. The main kinds of inference for state-space models. The shaded region is the interval for which we have data. The arrow represents the time step at which we want to perform inference. \(t\) is the current time, \(T\) is the sequence length, \(\ell\) is the lag and \(h\) is the prediction horizon.

Given the sequence of observations, and a known model, one of the main tasks with SSMs to perform posterior inference, about the hidden states; this is also called state estimation. At each time step \(t\), there are multiple forms of posterior we may be interested in computing, including the following:

- the filtering distribution \(p(\hmmhid_t|\hmmobs_{1:t})\)

- the smoothing distribution \(p(\hmmhid_t|\hmmobs_{1:T})\) (note that this conditions on future data \(T>t\))

- the fixed-lag smoothing distribution \(p(\hmmhid_{t-\ell}|\hmmobs_{1:t})\) (note that this infers \(\ell\) steps in the past given data up to the present).

We may also want to compute the predictive distribution \(h\) steps into the future:

(4)\[\begin{align} p(\hmmobs_{t+h}|\hmmobs_{1:t}) &= \sum_{\hmmhid_{t+h}} p(\hmmobs_{t+h}|\hmmhid_{t+h}) p(\hmmhid_{t+h}|\hmmobs_{1:t}) \end{align}\]

where the hidden state predictive distribution is

(5)\[\begin{align} p(\hmmhid_{t+h}|\hmmobs_{1:t}) &= \sum_{\hmmhid_{t:t+h-1}} p(\hmmhid_t|\hmmobs_{1:t}) p(\hmmhid_{t+1}|\hmmhid_{t}) p(\hmmhid_{t+2}|\hmmhid_{t+1}) \cdots p(\hmmhid_{t+h}|\hmmhid_{t+h-1}) \end{align}\]

See \cref{fig:dbn_inf_problems} for a summary of these distributions.

In addition to comuting posterior marginals, we may want to compute the most probable hidden sequence, i.e., the joint MAP estimate

\[\arg \max_{\hmmhid_{1:T}} p(\hmmhid_{1:T}|\hmmobs_{1:T})\]

or sample sequences from the posterior

\[\hmmhid_{1:T} \sim p(\hmmhid_{1:T}|\hmmobs_{1:T})\]

Algorithms for all these task are discussed in the following chapters, since the details depend on the form of the SSM.

## Example: inference in the casino HMM

We now illustrate filtering, smoothing and MAP decoding applied to the casino HMM from sec:casino.

    # Call inference engine

    filtered_dist, _, smoothed_dist, loglik = hmm.forward_backward(x_hist)
    map_path = hmm.viterbi(x_hist)

    /opt/anaconda3/lib/python3.8/site-packages/jax/_src/numpy/lax_numpy.py:5154: UserWarning: Explicitly requested dtype <class 'jax._src.numpy.lax_numpy.int64'> requested in astype is not available, and will be truncated to dtype int32. To enable more dtypes, set the jax_enable_x64 configuration option or the JAX_ENABLE_X64 shell environment variable. See https://github.com/google/jax#current-gotchas for more.
      lax_internal._check_user_dtype_supported(dtype, "astype")

    # Find the span of timesteps that the    simulated systems turns to be in state 1
    def find_dishonest_intervals(z_hist):
        spans = []
        x_init = 0
        for t, _ in enumerate(z_hist[:-1]):
            if z_hist[t + 1] == 0 and z_hist[t] == 1:
                x_end = t
                spans.append((x_init, x_end))
            elif z_hist[t + 1] == 1 and z_hist[t] == 0:
                x_init = t + 1
        return spans

    # Plot posterior
    def plot_inference(inference_values, z_hist, ax, state=1, map_estimate=False):
        n_samples = len(inference_values)
        xspan = np.arange(1, n_samples + 1)
        spans = find_dishonest_intervals(z_hist)
        if map_estimate:
            ax.step(xspan, inference_values, where="post")
        else:
            ax.plot(xspan, inference_values[:, state])

        for span in spans:
            ax.axvspan(*span, alpha=0.5, facecolor="tab:gray", edgecolor="none")
        ax.set_xlim(1, n_samples)
        # ax.set_ylim(0, 1)
        ax.set_ylim(-0.1, 1.1)
        ax.set_xlabel("Observation number")

     # Filtering
    fig, ax = plt.subplots()
    plot_inference(filtered_dist, z_hist, ax)
    ax.set_ylabel("p(loaded)")
    ax.set_title("Filtered")

    Text(0.5, 1.0, 'Filtered')

    # Smoothing
    fig, ax = plt.subplots()
    plot_inference(smoothed_dist, z_hist, ax)
    ax.set_ylabel("p(loaded)")
    ax.set_title("Smoothed")

    Text(0.5, 1.0, 'Smoothed')

    # MAP estimation
    fig, ax = plt.subplots()
    plot_inference(map_path, z_hist, ax, map_estimate=True)
    ax.set_ylabel("MAP state")
    ax.set_title("Viterbi")

    Text(0.5, 1.0, 'Viterbi')

    # TODO: posterior samples

## Example: inference in the tracking SSM

We now illustrate filtering, smoothing and MAP decoding applied to the 2d tracking HMM from Example: model for 2d tracking.

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Inference in discrete SSMs