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

摘要

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