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April 2016 On the convergence of adaptive sequential Monte Carlo methods
Alexandros Beskos, Ajay Jasra, Nikolas Kantas, Alexandre Thiery
Ann. Appl. Probab. 26(2): 1111-1146 (April 2016). DOI: 10.1214/15-AAP1113

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.

Citation

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Alexandros Beskos. Ajay Jasra. Nikolas Kantas. Alexandre Thiery. "On the convergence of adaptive sequential Monte Carlo methods." Ann. Appl. Probab. 26 (2) 1111 - 1146, April 2016. https://doi.org/10.1214/15-AAP1113

Information

Received: 1 February 2014; Revised: 1 January 2015; Published: April 2016
First available in Project Euclid: 22 March 2016

zbMATH: 1342.82127
MathSciNet: MR3476634
Digital Object Identifier: 10.1214/15-AAP1113

Subjects:
Primary: 60K35, 82C80
Secondary: 60F99, 62F15

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.26 • No. 2 • April 2016
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