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Limit theorems for sequential MCMC methods

Part of: Parametric inference Limit theorems Markov processes Probabilistic methods, simulation and stochastic differential equations Stochastic processes

Published online by Cambridge University Press:  15 July 2020

Axel Finke ,
Arnaud Doucet  and
Adam M. Johansen
Axel Finke*
Affiliation:
National University of Singapore
Arnaud Doucet*
Affiliation:
University of Oxford
Adam M. Johansen*
Affiliation:
University of Warwick & The Alan Turing Institute
*
*Postal address: Department of Statistics & Applied Probability, Block S16, Level 7, 6 Science Drive 2, Faculty of Science, National University of Singapore, Singapore 117546. Email: axel.finke@nus.edu.sg
*Postal address: Department of Statistics & Applied Probability, Block S16, Level 7, 6 Science Drive 2, Faculty of Science, National University of Singapore, Singapore 117546. Email: axel.finke@nus.edu.sg
***Postal address: Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK; The Alan Turing Institute, 96 Euston Rd, Kings Cross, London NW1 2DB, UK
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Abstract

Both sequential Monte Carlo (SMC) methods (a.k.a. ‘particle filters’) and sequential Markov chain Monte Carlo (sequential MCMC) methods constitute classes of algorithms which can be used to approximate expectations with respect to (a sequence of) probability distributions and their normalising constants. While SMC methods sample particles conditionally independently at each time step, sequential MCMC methods sample particles according to a Markov chain Monte Carlo (MCMC) kernel. Introduced over twenty years ago in [6], sequential MCMC methods have attracted renewed interest recently as they empirically outperform SMC methods in some applications. We establish an $\mathbb{L}_r$-inequality (which implies a strong law of large numbers) and a central limit theorem for sequential MCMC methods and provide conditions under which errors can be controlled uniformly in time. In the context of state-space models, we also provide conditions under which sequential MCMC methods can indeed outperform standard SMC methods in terms of asymptotic variance of the corresponding Monte Carlo estimators.

Keywords

Sequential Monte Carlo methodsparticle filtersMarkov chain Monte Carlo methodsalmost sure convergencemultivariate central limit theoremtime-uniform convergence

MSC classification

Primary: 65C05: Monte Carlo methods 60F05: Central limit and other weak theorems 60F25: $L^p$-limit theorems 60G35: Signal detection and filtering
Secondary: 60J05: Discrete-time Markov processes on general state spaces 62F15: Bayesian inference 68U20: Simulation

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 2 , June 2020 , pp. 377 - 403
Copyright
© Applied Probability Trust 2020

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