Blocking Strategies and Stability of Particle Gibbs Samplers

TL;DR

This paper investigates blocking strategies for Particle Gibbs samplers in Hidden Markov Models, aiming to achieve linear computational cost per iteration while maintaining stability and good mixing properties.

Contribution

It introduces and proves the stability of blocking strategies that are parallelizable and have linear cost growth, improving efficiency over existing methods.

Findings

Blocking strategies can be implemented with linear cost per iteration.

The proposed methods maintain the stability and mixing rate of the sampler.

Parallelizable algorithms enhance practical applicability for long sequences.

Abstract

Sampling from the conditional (or posterior) probability distribution of the latent states of a Hidden Markov Model, given the realization of the observed process, is a non-trivial problem in the context of Markov Chain Monte Carlo. To do this Andrieu et al. (2010) constructed a Markov kernel which leaves this conditional distribution invariant using a Particle Filter. From a practitioner's point of view, this Markov kernel attempts to mimic the act of sampling all the latent state variables as one block from the posterior distribution but for models where exact simulation is not possible. There are some recent theoretical results that establish the uniform ergodicity of this Markov kernel and that the mixing rate does not diminish provided the number of particles grows at least linearly with the number of latent states in the posterior. This gives rise to a cost, per application of the kernel, that is quadratic in the number of latent states which could be prohibitive for long observation sequences. We seek to answer an obvious but important question: is there a different implementation with a cost per-iteration that grows linearly with the number of latent states, but which is still stable in the sense that its mixing rate does not deteriorate? We address this problem using blocking strategies, which are easily parallelizable, and prove stability of the resulting sampler.