The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 26, Number 2 (2016), 1111-1146.
On the convergence of adaptive sequential Monte Carlo methods
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.
Ann. Appl. Probab., Volume 26, Number 2 (2016), 1111-1146.
Received: February 2014
Revised: January 2015
First available in Project Euclid: 22 March 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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