Limit theorems for sequential MCMC methods

Abstract

Sequential Monte Carlo (SMC) methods, also known as particle filters (PFs), 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 Markov chain Monte Carlo (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.

Invertible Particle-Flow-Based Sequential MCMC With Extension to Gaussian Mixture Noise Models

“…This paper derives the asymptotic convergence results of the proposed algorithms. Finite sample analysis of filter errors is an important direction to explore; [36] provides a valuable finite sample bound for SMCMC errors. In a similar manner to [46], it may be possible to identify a relationship between the magnitude of the error and the stiffness of the flow.…”

“…While similar in spirit to results presented in [35] for the SIMCMC algorithm, the key assumptions are slightly less restrictive and the theorem directly addresses the sequential implementation in [33] and this work. Recently, [36] carries out a rigorous statistical analysis of SMCMC algorithms to establish upper bounds on finite sample filter errors, in addition to providing asymptotic convergence results.…”

An invitation to sequential Monte Carlo samplers

“…This shortcoming of SMC has attracted some attention and variants exist where new particles can be added along the way, e.g. see Brockwell et al [2010], Paige et al [2014], Murray et al [2016b], Finke et al [2018].…”