The Annals of Statistics

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

Nicolas Chopin

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Abstract

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 (πt). 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 πt, 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.

Article information

Source
Ann. Statist. Volume 32, Number 6 (2004), 2385-2411.

Dates
First available in Project Euclid: 7 February 2005

Permanent link to this document
http://projecteuclid.org/euclid.aos/1107794873

Digital Object Identifier
doi:10.1214/009053604000000698

Mathematical Reviews number (MathSciNet)
MR2153989

Zentralblatt MATH identifier
1079.65006

Subjects
Primary: 65C05: Monte Carlo methods 62F15: Bayesian inference 60F05: Central limit and other weak theorems
Secondary: 82C80: Numerical methods (Monte Carlo, series resummation, etc.) 62L10: Sequential analysis

Keywords
Markov chain Monte Carlo particle filter recursive Monte Carlo filter resample-move algorithms residual resampling state-space model

Citation

Chopin, Nicolas. Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 (2004), no. 6, 2385--2411. doi:10.1214/009053604000000698. http://projecteuclid.org/euclid.aos/1107794873.


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