Central Limit Theorem for Sequential Monte Carlo methods and its Applications to Bayesian Inference

• Activity Number: 199
• Type: Topic Contributed
• Date/Time: Tuesday, August 5, 2003 : 8:30 AM to 10:20 AM
• Sponsor: Section on Bayesian Stat. Sciences
• Author(s): Nicolas Chopin*+
• Companies: ENSAE
• Keywords: Monte Carlo Markov chain ; particle filter ; recursive Monte Carlo filter ; resample-move algorithms ; residual resampling ; state-space model

Abstract

The terms "Sequential Monte Carlo methods," or similarly "particle filters," refer to a general class of iterative algorithms which perform Monte Carlo approximations of a given sequence of distributions of interest pi_t. Their use is usually justified by first-order asymptotics, i.e. it is shown that computed estimates converge almost surely as the number of "particles" (simulated values) tends towards infinity. We establish a central limit theorem for these estimates. This result holds under minimal assumptions on the distributions pi_t, and apply in a general framework which encompasses most of sequential Monte Carlo methods that have been considered in the literature, including the resample-move algorithm of Gilks and Berzuini (2001) or the residual resampling scheme of Liu and Chen (1998). 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 to which rate these asymptotic variances diverge in time, in order to assess the long term reliability of the considered algorithm.