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State-space models and Feynman-Kac representations

This section gives a (very brief) introduction to state-space models and their Feynman-Kac representations, and introduces some notation used in the documentation of GenericSSMs.jl.

State-space models

State-space models (SSMs) are a broad class of statistical models for modelling multivariate time series data. Suppose that we are interested in modelling a $p$-dimensional time series of $n$ observations, $y_{1:n} = y_1, y_2, \ldots, y_n$. SSMs assume that $y_{1:n}$ are generated conditional on a latent (unknown) state process, $x_{1:n} = x_1, x_2, \ldots, x_n$ whose dynamics form a Markov process. A state-space model can thus be written in the following form:

\[ \begin{aligned} \text{State process/equation:} \\ x_1 &\sim m_1(\cdot) \\ x_k &\sim m_k(\cdot \mid x_{k-1}) \ \text{ for } k \geq 2 \\ \text{Observation process/equation:} \\ y_k &\sim g_k(\cdot \mid x_k) \ \text{ for } k \geq 1 \\ \end{aligned}\]

where:

  • $m_1$ is the distribution of the first latent state, $x_1$.
  • $m_k(\cdot \mid x_{k-1})$ for $k \geq 2$ are Markov transitions of the latent state, and
  • $g_k(y_k \mid x_k)$ for $k \geq 1$ are densities of the observations $y_k$ given the state $x_k$.

In summary, the probability distributions $m_k, k \geq 1$ constitute the dynamics of the latent state process $x_{1:n}$, and $g_k, k \geq 1$ describe how the (known) observations are generated conditional on the state process. State-space modelling means designing the state variables $x_{1:n}$, their dynamics $m_{1:n}$ and the observation densities $g_{1:n}$ such that the model comprised of these provides a sensible statistical model for the observed data $y_{1:n}$.

Computational problem

Assume for simplicity that $m_k, k \geq 1$ admit densities (although this assumption is necessary only for some of the functionality of GenericSSMs). Then, the joint density of the states $x_{1:n}$ and observations $y_{1:n}$ is given by:

\[ \pi(x_{1:n}, y_{1:n}) = m_1(x_1)g_1(y_1 \mid x_1)\prod_{k = 2}^{n} m_k(x_k \mid x_{k-1}) g_k(y_k \mid x_k).\]

In the context of SSMs, the central computational problem is in the computation of various posterior distributions of the unknown latent states $x_{1:n}$. Indeed, GenericSSMs provides functionality for simulating from the following posterior distributions:

  • $p(x_{1:k} \mid y_{1:k}) \text{ for some } k \geq 1$ (filtering distributions)

  • $p(x_{n + h} \mid y_{1:n})$ or $p(y_{n + h} \mid y_{1:n})$ for some $h > 0$ (predictive distributions)

  • $p(x_{1:n} \mid y_{1:n})$ (smoothing distribution)

Furthermore, the particle filter also gives access to an estimate of the normalising constant, $p(y_{1:n})$.

Feynman-Kac representation of an SSM

In GenericSSMs.jl, SSMs are defined in terms of Feynman-Kac models [see Del Moral (2004), Chopin and Papaspiliopoulos (2020)] that are alternative representations for SSMs. There are multiple possible Feynman-Kac models for a single (underlying) SSM. The rationale for using Feynman-Kac models is that they provide a convenient abstraction over SSMs that is very well suited for generic programming.

A Feynman-Kac model expresses

\[ \pi(x_{1:n} \mid y_{1:n}) \propto m_1(x_1)g_1(y_1 \mid x_1)\prod_{k = 2}^{n} m_k(x_k \mid x_{k-1}) g_k(y_k \mid x_k)\]

of the underlying SSM $(m_{1:n}, g_{1:n})$ in terms of `components` $(M_{1:n}, G_{1:n})$, such that it is assumed that

\[ \pi(x_{1:n} \mid y_{1:n}) \propto M_1(x_1)G_1(x_1) \prod_{k = 2}^{n} M_k(x_k \mid x_{k-1}) G_k(x_{k-1}, x_k) \ \text{ for all } x_{1:n}.\]

The components $(M_{1:n}, G_{1:n})$ consists of:

  • $M_1$, which is an (alternative) initial distribution for $x_1$
  • $M_k$ for $k \geq 2$ that are (alternative) Markov transitions of the state, and
  • $G_k$ for $k \geq 1$ that are `potential functions` taking values in the non-negative reals.

In other words, the previous equation simply says that the joint distribution of a Feynman-Kac model must equal the joint posterior of the underlying SSM (up to a constant of proportionality).

Note that in the above we have again assumed that $M_k$ admit densities (although this is not required by all algorithms of GenericSSMs.jl, see Method definitions required by use case). In general, $M_k$ should however be simulatable, and $G_k$ should be evaluatable pointwise.

An example of a Feynman-Kac model $(M_{1:n}, G_{1:n})$ for the SSM $(m_{1:n}, g_{1:n})$ is given by the choice

\[ \begin{aligned} M_k &:= q_k \\ G_k &:= \dfrac{g_k m_k}{q_k}, \\ \end{aligned}\]

where $q_k$ is a `proposal distribution` for the states. A special case of this Feynman-Kac model is the choice $q_k = m_k$, which yields the bootstrap filter of [Gordon, Salmond, Smith (1993)].

References
  • Del Moral, P. (2004) Feynman-Kac Formulae. Springer.
  • Chopin, N., and Papaspiliopoulos, O. (2020) An introduction to sequential Monte Carlo. Springer.
  • Gordon, N. J., Salmond, D. J., and Smith, A. F. (1993) Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F (Radar and Signal Processing). 140(2):107-113.