Coupled conditional backward sampling particle filter

TL;DR

This paper introduces a coupled conditional backward sampling particle filter (CCBPF) that remains effective with a fixed number of particles regardless of the time horizon, providing theoretical stability and practical unbiased estimation benefits.

Contribution

The paper presents a new coupling analysis of CBPF, demonstrating its stability and effectiveness for fixed particles over long time series, and introduces CCBPF for unbiased smoothing estimation.

Findings

CCBPF coupling time grows linearly with time horizon.

Fixed particle number suffices for effective long-term smoothing.

Unbiased estimation enables confidence intervals and parallel computing.

Abstract

The conditional particle filter (CPF) is a promising algorithm for general hidden Markov model smoothing. Empirical evidence suggests that the variant of CPF with backward sampling (CBPF) performs well even with long time series. Previous theoretical results have not been able to demonstrate the improvement brought by backward sampling, whereas we provide rates showing that CBPF can remain effective with a fixed number of particles independent of the time horizon. Our result is based on analysis of a new coupling of two CBPFs, the coupled conditional backward sampling particle filter (CCBPF). We show that CCBPF has good stability properties in the sense that with fixed number of particles, the coupling time in terms of iterations increases only linearly with respect to the time horizon under a general (strong mixing) condition. The CCBPF is useful not only as a theoretical tool, but also as a practical method that allows for unbiased estimation of smoothing expectations, following the recent developments by Jacob et al. (to appear). Unbiased estimation has many advantages, such as enabling the construction of asymptotically exact confidence intervals and straightforward parallelisation.