The Bootstrap Particle Filtering Bias

Jimmy Olsson and Tobias Rydén

Abstract:: Particle filter methods constitute a class of iterative genetic-type algorithms which provide powerful tools for obtaining approximate solutions to non-linear and/or non-Gaussian filtering problems. The aim of this paper is to, using standard tools from probability theory, study the bias of Monte Carlo integration estimates obtained by the bootstrap particle filter. A bound on this bias, which is geometrically growing in time and inversely proportional to the number N of particles of the system, is derived. Under suitable mixing assumptions on the latent Markov model, a bound of the bias which is uniform with respect to the time parameter and inversely proportional to N is obtained. In the last part of the paper we investigate the behaviour of the bias as N goes to infinity; it will be seen that the bias, for a fixed time point, is indeed asymptotically inversely proportional to N.

Key words:: Bootstrap particle filter, interacting particle system, non-linear filtering, sequential Monte Carlo methods, state space model