Asymptotic genealogies of interacting particle systems with an application to sequential Monte Carlo

TL;DR

This paper analyzes the genealogical structures of sequential Monte Carlo methods, showing they converge to Kingman's coalescent under certain conditions, which helps in understanding and improving algorithm performance.

Contribution

It establishes conditions under which SMC genealogies converge to Kingman's coalescent, linking genealogical structure to algorithm efficiency and design.

Findings

SMC genealogies converge to Kingman's coalescent in the infinite system limit.

The paper characterizes the limiting mean and variance of the genealogical tree height.

Simulation suggests the convergence conditions may be less restrictive than initially stated.

Abstract

We study weighted particle systems in which new generations are resampled from current particles with probabilities proportional to their weights. This covers a broad class of sequential Monte Carlo (SMC) methods, widely-used in applied statistics and cognate disciplines. We consider the genealogical tree embedded into such particle systems, and identify conditions, as well as an appropriate time-scaling, under which they converge to the Kingman n-coalescent in the infinite system size limit in the sense of finite-dimensional distributions. Thus, the tractable n-coalescent can be used to predict the shape and size of SMC genealogies, as we illustrate by characterising the limiting mean and variance of the tree height. SMC genealogies are known to be connected to algorithm performance, so that our results are likely to have applications in the design of new methods as well. Our conditions for convergence are strong, but we show by simulation that they do not appear to be necessary.