Sequential Markov Chain Monte Carlo for Filtering of State-Space Models with Low or Degenerate Observation Noise

TL;DR

This paper develops sequential Markov chain Monte Carlo methods for filtering in state-space models with low or degenerate observation noise, providing theoretical convergence results and demonstrating effectiveness on complex models.

Contribution

It introduces a novel filtering approach using sequential MCMC tailored for low or degenerate noise scenarios, with convergence analysis and practical applications.

Findings

Sequential MCMC converges as noise diminishes in low-noise models.

The method accurately approximates filters on manifolds defined by the observation function.

Performance demonstrated on challenging stochastic models from statistics and applied mathematics.

Abstract

We consider the discrete-time filtering problem in scenarios where the observation noise is degenerate or low. More precisely, one is given access to a discrete time observation sequence which at any time $k$ depends only on the state of an unobserved Markov chain. We specifically assume that the functional relationship between observations and hidden Markov chain has either degenerate or low noise. In this article, under suitable assumptions, we derive the filtering density and its recursions for this class of problems on a specific sequence of manifolds defined through the observation function. We then design sequential Markov chain Monte Carlo methods to approximate the filter serially in time. For a certain linear observation model, we show that using sequential Markov chain Monte Carlo for low noise will converge as the noise disappears to that of using sequential Markov chain Monte Carlo for degenerate noise. We illustrate the performance of our methodology on several challenging stochastic models deriving from Statistics and Applied Mathematics.