Sequential Monte Carlo smoothing for general state space hidden Markov models

TL;DR

This paper develops a particle-based framework for approximating smoothing distributions in general hidden Markov models, providing theoretical convergence guarantees and an efficient linear-cost algorithm.

Contribution

It introduces a unifying analysis of particle smoothing schemes, establishing convergence results and proposing a computationally efficient joint smoothing algorithm.

Findings

Proves exponential deviation inequalities for particle smoothers.

Establishes time-uniform bounds on smoothing errors under mixing conditions.

Proposes a linear-cost algorithm for joint smoothing distribution approximation.

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.