Particle-based, rapid incremental smoother meets particle Gibbs

TL;DR

This paper introduces the Parisian particle Gibbs (PPG) sampler, a novel method that reduces bias in particle-based online smoothing by wrapping the PARIS algorithm within a particle Gibbs framework, supported by theoretical and empirical validation.

Contribution

It proposes the PPG sampler, significantly reducing bias in particle smoothing while maintaining similar complexity, and provides theoretical bounds and numerical experiments.

Findings

PPG reduces bias compared to PARIS for a given computational cost.

Theoretical bounds on bias, variance, and deviation inequalities are established.

Numerical experiments support the effectiveness of the PPG method.

Abstract

The particle-based, rapid incremental smoother (PARIS) is a sequential Monte Carlo technique allowing for efficient online approximation of expectations of additive functionals under Feynman--Kac path distributions. Under weak assumptions, the algorithm has linear computational complexity and limited memory requirements. It also comes with a number of non-asymptotic bounds and convergence results. However, being based on self-normalised importance sampling, the PARIS estimator is biased; its bias is inversely proportional to the number of particles but has been found to grow linearly with the time horizon under appropriate mixing conditions. In this work, we propose the Parisian particle Gibbs (PPG) sampler, whose complexity is essentially the same as that of the PARIS and which significantly reduces the bias for a given computational complexity at the price of a modest increase in the variance. This method is a wrapper in the sense that it uses the PARIS algorithm in the inner loop of particle Gibbs to form a bias-reduced version of the targeted quantities. We substantiate the PPG algorithm with theoretical results, including new bounds on bias and variance as well as deviation inequalities. We illustrate our theoretical results with numerical experiments supporting our claims.