Long-term stability of sequential Monte Carlo methods under verifiable conditions

TL;DR

This paper proves that bootstrap particle filters maintain long-term stability with bounded variance and error under mild conditions, even with noncompact state spaces and model misspecification.

Contribution

It establishes novel theoretical bounds on the long-term stability of bootstrap particle filters under very mild assumptions, including noncompact state spaces and model misspecification.

Findings

Asymptotic variance is uniformly bounded in time.

Time uniform bounds on asymptotic L^p error are derived.

Results apply to misspecified models and noncompact state spaces.

Abstract

This paper discusses particle filtering in general hidden Markov models (HMMs) and presents novel theoretical results on the long-term stability of bootstrap-type particle filters. More specifically, we establish that the asymptotic variance of the Monte Carlo estimates produced by the bootstrap filter is uniformly bounded in time. On the contrary to most previous results of this type, which in general presuppose that the state space of the hidden state process is compact (an assumption that is rarely satisfied in practice), our very mild assumptions are satisfied for a large class of HMMs with possibly noncompact state space. In addition, we derive a similar time uniform bound on the asymptotic $\mathsf{L}^p$ error. Importantly, our results hold for misspecified models; that is, we do not at all assume that the data entering into the particle filter originate from the model governing the dynamics of the particles or not even from an HMM.