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Limit theorems for sequential MCMC methods

Published online by Cambridge University Press:  15 July 2020

Axel Finke
Affiliation:
National University of Singapore
Arnaud Doucet
Affiliation:
University of Oxford
Adam M. Johansen
Affiliation:
University of Warwick & The Alan Turing Institute
Corresponding
E-mail address:

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.

Type
Original Article
Copyright
© Applied Probability Trust 2020

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