Limit theorems for sequential MCMC methods

TL;DR

This paper establishes theoretical limit theorems for sequential MCMC methods, demonstrating conditions under which they outperform traditional SMC methods in terms of error control and asymptotic variance.

Contribution

It provides the first strong law of large numbers and central limit theorem for sequential MCMC, with conditions for uniform error control and improved performance in state-space models.

Findings

Proves strong law of large numbers for sequential MCMC

Establishes central limit theorem for these methods

Shows conditions where sequential MCMC outperforms SMC in variance reduction

Abstract

Sequential Monte Carlo (SMC) methods, also known as particle filters, constitute a class of algorithms used to approximate expectations with respect to a sequence of probability distributions as well as the normalising constants of those distributions. Sequential MCMC methods are an alternative class of techniques addressing similar problems in which particles are sampled according to an MCMC kernel rather than conditionally independently at each time step. These methods were introduced over twenty years ago by Berzuini et al. (1997). Recently, there has been a renewed interest in such algorithms as they demonstrate an empirical performance superior to that of SMC methods in some applications. We establish a strong law of large numbers and a central limit theorem for sequential MCMC methods and provide conditions under which errors can be controlled uniformly in time. In the context of state-space models, we provide conditions under which sequential MCMC methods can indeed outperform standard SMC methods in terms of asymptotic variance of the corresponding Monte Carlo estimators.