# Background

A discrete-time, Markovian state space model takes the form \begin{align} \mathbf{y}_t&\sim p(\mathbf{y}_t\,|\,\mathbf{s}_t,\,\boldsymbol{\theta})\\ \mathbf{s}_t&\sim p(\mathbf{s}_t\,|\,\mathbf{s}_{t-1},\,\boldsymbol{\theta})\\ \mathbf{s}_0&\sim p(\mathbf{s}_0\,|\,\boldsymbol{\theta}), \end{align} where the $$\mathbf{y}_t$$ are observable, the $$\mathbf{s}_t$$ are latent, and $$\boldsymbol{\theta}$$ collects all of the parameters (assumed known) that govern the system.

To solve the filtering problem, we want to recursively compute the marginal posteriors $$p(\mathbf{s}_t\,|\,\mathbf{y}_{1:t},\,\boldsymbol{\theta}) = \frac{p(\mathbf{y}_t\,|\,\mathbf{s}_t,\,\boldsymbol{\theta})p(\mathbf{s}_t\,|\,\mathbf{y}_{1:t-1},\,\boldsymbol{\theta})}{p(\mathbf{y}_t\,|\,\mathbf{y}_{1:t-1},\,\boldsymbol{\theta})}$$ known as the filtering distributions. This requires that we be able to compute the predictive distributions \begin{align} p(\mathbf{s}_t\,|\,\mathbf{y}_{1:t-1},\,\boldsymbol{\theta})&=\int p(\mathbf{s}_t\,|\,\mathbf{s}_{t-1},\,\mathbf{y}_{1:t-1},\,\boldsymbol{\theta})p(\mathbf{s}_{t-1}\,|\,\mathbf{y}_{1:t-1},\,\boldsymbol{\theta}) \,\textrm{d}\mathbf{s}_{t-1} \\ p(\mathbf{y}_t\,|\,\mathbf{y}_{1:t-1},\,\boldsymbol{\theta})&=\int p(\mathbf{y}_t\,|\,\mathbf{s}_t,\,\boldsymbol{\theta})p(\mathbf{s}_t\,|\,\mathbf{y}_{1:t-1},\,\boldsymbol{\theta}) \,\textrm{d}\mathbf{s}_{t}. \end{align} There is work that I don't understand on necessary and sufficient conditions for when these integrals are tractable, but I don't think we have a single set of necessary and sufficient conditions that lock the problem down (please correct me if I'm wrong!).

# Examples

Closed-form solutions are known in two famous examples.

### Hidden Markov Model

In this model,

\begin{align} \mathbf{y}_t&\sim p(\mathbf{y}_t\,|\,s_t,\,\boldsymbol{\phi})\\ s_t&\sim \text{Categorical}(\mathbf{p}_{s_{t-1}})\\ s_0&\sim \text{Categorical}(\mathbf{p}_0), \end{align} where $$\boldsymbol{\theta} = \left\{\boldsymbol{\phi},\,\{\mathbf{p}_j\}_{j=0}^M\right\}$$. So the measurement distribution can be whatever, and the univariate latent state follows a discrete-state Markov chain. In this case, the integrals become finite sums.

### Linear, Gaussian System

In this model,

\begin{align} \mathbf{y}_t&=\mathbf{A}\mathbf{s}_t + \mathbf{b}+\boldsymbol{\varepsilon}_t, &&\boldsymbol{\varepsilon}_t\sim N(\mathbf{0},\,\mathbf{\Sigma})\\ \mathbf{s}_t&=\mathbf{C}\mathbf{s}_{t-1} + \mathbf{d}+\boldsymbol{\eta}_t, &&\boldsymbol{\eta}_t\sim N(\mathbf{0},\,\mathbf{\Omega})\\ \mathbf{s}_0&\sim N(\bar{\mathbf{s}}_{0},\,\mathbf{P}_0), \end{align} where $$\boldsymbol{\theta}=\{\mathbf{A},\,\mathbf{b},\,\boldsymbol{\Sigma},\,\mathbf{C},\,\mathbf{d},\,\boldsymbol{\Omega},\,\bar{\mathbf{s}}_{0},\,\mathbf{P}_0\}$$. In this case, all of the predictive and filtering distributions are Gaussian, and their moments can be computed using the Kalman filter.

### Others

In addition to these, we have analytical solutions for

# Question

What are other examples of state space models where the filtering problem can be solved analytically? I ask with the hope of collecting in one place a running list of examples.