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On the convergence of adaptive sequential Monte Carlo methods

Beskos, A; Jasra, A; Kantas, N; Thiery, A; (2016) On the convergence of adaptive sequential Monte Carlo methods. Annals of Applied Probability , 26 (2) pp. 1111-1146. 10.1214/15-AAP1113. Green open access

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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 highdimensional parameters of the proposal kernels is critical for the efficiency of the algorithm.

Type: Article
Title: On the convergence of adaptive sequential Monte Carlo methods
Open access status: An open access version is available from UCL Discovery
DOI: 10.1214/15-AAP1113
Publisher version: http://dx.doi.org/10.1214/15-AAP1113
Language: English
Additional information: Copyright © Institute of Mathematical Statistics, 2016.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery.ucl.ac.uk/id/eprint/1400010
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