# 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.

## 1 Answer

For a list of examples, in the thesis Bayesian state estimation in partially observable Markov processes (available here), the author explores many different models summed up in Figure 1 (in French) in page 12. Then in page 25 of the document, he claims, with appropriate references, that (C6) (C5) (D2) (D1) (H2) are models where the filtering problem can always be solved exactly.