Variance estimation in the particle filter

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 Monte Carlo approximations that particle filters deliver. All our estimators can be computed from a single run of a particle filter. 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 estimate each of these terms consistently. When the number of particles may vary over time, this allows approximation of the asymptotically optimal allocation of particle numbers.