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

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.

Published in at http://dx.doi.org/10.1214/13-AAP962 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Recursive Monte Carlo filters: Algorithms and theoretical analysis
• Sequential Monte Carlo smoothing for general state space hidden Markov models
• Asymptotic properties of the maximum likelihood estimation in misspecified hidden Markov models
• The stability of conditional Markov processes and Markov chains in random environments
• Moderate deviations for particle filtering

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• Approximate Smoothing and Parameter Estimation in High-Dimensional State-Space Models
• Limit theorems for sequential MCMC methods
• Particle-based adaptive-lag online marginal smoothing in general state-space models