A pseudo-marginal sequential Monte Carlo online smoothing algorithm

TL;DR

This paper introduces an extended online smoothing algorithm using pseudo-marginal techniques that efficiently handles intractable transition densities in complex hidden Markov models, with proven convergence and stability.

Contribution

It extends the PaRIS algorithm to intractable transition densities, providing a linear complexity, constant memory, and theoretical guarantees including a bias bound.

Findings

Algorithm has linear complexity in particles

Proven convergence with a central limit theorem

Bias bound of O(n ε) under strong mixing

Abstract

We consider online computation of expectations of additive state functionals under general path probability measures proportional to products of unnormalised transition densities. These transition densities are assumed to be intractable but possible to estimate, with or without bias. Using pseudo-marginalisation techniques we are able to extend the particle-based, rapid incremental smoother (PaRIS) algorithm proposed in [J.Olsson and J.Westerborn. Efficient particle-based online smoothing in general hidden Markov models: The PaRIS algorithm. Bernoulli, 23(3):1951--1996, 2017] to this setting. The resulting algorithm, which has a linear complexity in the number of particles and constant memory requirements, applies to a wide range of challenging path-space Monte Carlo problems, including smoothing in partially observed diffusion processes and models with intractable likelihood. The algorithm is furnished with several theoretical results, including a central limit theorem, establishing its convergence and numerical stability. Moreover, under strong mixing assumptions we establish a novel $O(n \varepsilon)$ bound on the asymptotic bias of the algorithm, where $n$ is the path length and $\varepsilon$ controls the bias of the density estimators.