On resampling schemes for particle filters with weakly informative observations

TL;DR

This paper analyzes different resampling schemes in particle filters with weakly informative observations, especially for continuous-time models, showing that some schemes have lower resampling rates and variance, leading to more efficient filtering.

Contribution

It introduces a continuous-time limit analysis of resampling schemes and proves convergence of particle approximations to a continuous-time system under general conditions.

Findings

Systematic and SSP resampling dominate stratified and independent resampling in limiting resampling rate.

Reduced resampling intensity leads to lower variance in particle filter estimates.

The particle approximation converges to a continuous-time particle system as discretization becomes finer.

Abstract

We consider particle filters with weakly informative observations (or `potentials') relative to the latent state dynamics. The particular focus of this work is on particle filters to approximate time-discretisations of continuous-time Feynman--Kac path integral models -- a scenario that naturally arises when addressing filtering and smoothing problems in continuous time -- but our findings are indicative about weakly informative settings beyond this context too. We study the performance of different resampling schemes, such as systematic resampling, SSP (Srinivasan sampling process) and stratified resampling, as the time-discretisation becomes finer and also identify their continuous-time limit, which is expressed as a suitably defined `infinitesimal generator.' By contrasting these generators, we find that (certain modifications of) systematic and SSP resampling `dominate' stratified and independent `killing' resampling in terms of their limiting overall resampling rate. The reduced intensity of resampling manifests itself in lower variance in our numerical experiment. This efficiency result, through an ordering of the resampling rate, is new to the literature. The second major contribution of this work concerns the analysis of the limiting behaviour of the entire population of particles of the particle filter as the time discretisation becomes finer. We provide the first proof, under general conditions, that the particle approximation of the discretised continuous-time Feynman--Kac path integral models converges to a (uniformly weighted) continuous-time particle system.