Simple conditions for convergence of sequential Monte Carlo genealogies with applications

TL;DR

This paper establishes simple conditions under which the genealogical process of certain particle systems converges to a Kingman coalescent, with applications to sequential Monte Carlo algorithms.

Contribution

It provides new, straightforward criteria for genealogical convergence in particle systems, applicable to common SMC algorithms and resampling schemes.

Findings

Conditions verified for standard SMC algorithms with low-variance resampling

Conditions verified for conditional SMC with multinomial resampling

Genealogical process converges to a Kingman coalescent under these conditions

Abstract

We present simple conditions under which the limiting genealogical process associated with a class of interacting particle systems with non-neutral selection mechanisms, as the number of particles grows, is a time-rescaled Kingman coalescent. Sequential Monte Carlo algorithms are popular methods for approximating integrals in problems such as non-linear filtering and smoothing which employ this type of particle system. Their performance depends strongly on the properties of the induced genealogical process. We verify the conditions of our main result for standard sequential Monte Carlo algorithms with a broad class of low-variance resampling schemes, as well as for conditional sequential Monte Carlo with multinomial resampling.