Sequential Monte Carlo smoothing for general state space hidden Markov models

Abstract

Computing smoothing distributions, the distributions of one or more states conditional on past, present, and future observations is a recurring problem when operating on general hidden Markov models. The aim of this paper is to provide a foundation of particle-based approximation of such distributions and to analyze, in a common unifying framework, different schemes producing such approximations. In this setting, general convergence results, including exponential deviation inequalities and central limit theorems, are established. In particular, time uniform bounds on the marginal smoothing error are obtained under appropriate mixing conditions on the transition kernel of the latent chain. In addition, we propose an algorithm approximating the joint smoothing distribution at a cost that grows only linearly with the number of particles.

Published in at http://dx.doi.org/10.1214/10-AAP735 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1012.4183 by other authors

References in corpus (5)

• Recursive Monte Carlo filters: Algorithms and theoretical analysis
• Uniform observability of hidden Markov models and filter stability for unstable signals
• Uniform Time Average Consistency of Monte Carlo Particle Filters
• The stability of conditional Markov processes and Markov chains in random environments
• Stability of nonlinear filters in nonmixing case

Cited by in corpus (8)

• Three discussions of the paper "sequential quasi-Monte Carlo sampling", by M. Gerber and N. Chopin
• Particle approximation improvement of the joint smoothing distribution with on-the-fly variance estimation
• Efficient particle-based online smoothing in general hidden Markov models: the PaRIS algorithm
• Approximate Bayesian Computation for Smoothing
• A Sharp First Order Analysis of Feynman-Kac Particle Models
• Metropolising forward particle filtering backward sampling and Rao-Blackwellisation of Metropolised particle smoothers
• Convergence of Sequential Quasi-Monte Carlo Smoothing Algorithms
• Particle-based likelihood inference in partially observed diffusion processes using generalised Poisson estimators