Variance estimation in the particle filter

TL;DR

This paper introduces a method to estimate Monte Carlo errors in particle filters by tracking genealogical structures, enabling variance estimation from a single run without additional simulations.

Contribution

It presents novel variance estimators based on genealogical tracking that are consistent and can be computed from a single particle filter run.

Findings

Estimators are weakly consistent for asymptotic variances.

Some estimators are non-asymptotically unbiased.

Method allows variance decomposition per time step.

Abstract

This paper concerns numerical assessment of Monte Carlo error in particle filters. We show that by keeping track of certain key features of the genealogical structure arising from resampling operations, it is possible to estimate variances of a number of standard Monte Carlo approximations which particle filters deliver. All our estimators can be computed from a single run of a particle filter with no further simulation. We establish that as the number of particles grows, our estimators are weakly consistent for asymptotic variances of the Monte Carlo approximations and some of them are also non-asymptotically unbiased. The asymptotic variances can be decomposed into terms corresponding to each time step of the algorithm, and we show how to consistently estimate each of these terms. When the number of particles may vary over time, this allows approximation of the asymptotically optimal allocation of particle numbers.