Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference

Details Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference

2005-08-30

arxiv.org/abs/math/0508594v1

Annals of Statistics 2004, Vol. 32, No. 6, 2385-2411

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053604000000698 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053604000000698

The term ``sequential Monte Carlo methods'' or, equivalently, ``particle filters,'' refers to a general class of iterative algorithms that performs Monte Carlo approximations of a given sequence of distributions of interest (\pi_t). We establish in this paper a central limit theorem for the Monte Carlo estimates produced by these computational methods. This result holds under minimal assumptions on the distributions \pi_t, and applies in a general framework which encompasses most of the sequential Monte Carlo methods that have been considered in the literature, including the resample-move algorithm of Gilks and Berzuini [J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 127-146] and the residual resampling scheme. The corresponding asymptotic variances provide a convenient measurement of the precision of a given particle filter. We study, in particular, in some typical examples of Bayesian applications, whether and at which rate these asymptotic variances diverge in time, in order to assess the long term reliability of the considered algorithm.

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