The Annals of Applied Probability

On the convergence of adaptive sequential Monte Carlo methods

Alexandros Beskos, Ajay Jasra, Nikolas Kantas, and Alexandre Thiery

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Abstract

In several implementations of Sequential Monte Carlo (SMC) methods it is natural and important, in terms of algorithmic efficiency, to exploit the information of the history of the samples to optimally tune their subsequent propagations. In this article we provide a carefully formulated asymptotic theory for a class of such adaptive SMC methods. The theoretical framework developed here will cover, under assumptions, several commonly used SMC algorithms [Chopin, Biometrika 89 (2002) 539–551; Jasra et al., Scand. J. Stat. 38 (2011) 1–22; Schäfer and Chopin, Stat. Comput. 23 (2013) 163–184]. There are only limited results about the theoretical underpinning of such adaptive methods: we will bridge this gap by providing a weak law of large numbers (WLLN) and a central limit theorem (CLT) for some of these algorithms. The latter seems to be the first result of its kind in the literature and provides a formal justification of algorithms used in many real data contexts [Jasra et al. (2011); Schäfer and Chopin (2013)]. We establish that for a general class of adaptive SMC algorithms [Chopin (2002)], the asymptotic variance of the estimators from the adaptive SMC method is identical to a “limiting” SMC algorithm which uses ideal proposal kernels. Our results are supported by application on a complex high-dimensional posterior distribution associated with the Navier–Stokes model, where adapting high-dimensional parameters of the proposal kernels is critical for the efficiency of the algorithm.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 2 (2016), 1111-1146.

Dates
Received: February 2014
Revised: January 2015
First available in Project Euclid: 22 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1458651829

Digital Object Identifier
doi:10.1214/15-AAP1113

Mathematical Reviews number (MathSciNet)
MR3476634

Zentralblatt MATH identifier
1342.82127

Subjects
Primary: 82C80: Numerical methods (Monte Carlo, series resummation, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F99: None of the above, but in this section 62F15: Bayesian inference

Keywords
Adaptive sequential Monte Carlo CLT MCMC

Citation

Beskos, Alexandros; Jasra, Ajay; Kantas, Nikolas; Thiery, Alexandre. On the convergence of adaptive sequential Monte Carlo methods. Ann. Appl. Probab. 26 (2016), no. 2, 1111--1146. doi:10.1214/15-AAP1113. https://projecteuclid.org/euclid.aoap/1458651829


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