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On the convergence rates of some adaptive Markov chain Monte Carlo algorithms

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  30 March 2016

Yves Atchadé  and
Yizao Wang
Yves Atchadé*
Affiliation:
University of Michigan, Ann Arbor
Yizao Wang*
Affiliation:
University of Cincinnati
*
∗ Postal address: Department of Statistics, University of Michigan, 439 West Hall, 1085 South University, Ann Arbor, MI 48109-1107, USA.
∗∗ Postal address: Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way, Cincinnati OH 45221, USA. Email address: yizao.wang@uc.edu
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Abstract

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In this paper we study the mixing time of certain adaptive Markov chain Monte Carlo (MCMC) algorithms. Under some regularity conditions, we show that the convergence rate of importance resampling MCMC algorithms, measured in terms of the total variation distance, is O(n-1). By means of an example, we establish that, in general, this algorithm does not converge at a faster rate. We also study the interacting tempering algorithm, a simplified version of the equi-energy sampler, and establish that its mixing time is of order O(n-1/2).

Keywords

Adaptive Markov chain Monte Carlomixing timetotal variation distanceimportance resampling algorithmequi-energy sampler

MSC classification

Primary: 65C05: Monte Carlo methods 65C40: Computational Markov chains
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 811 - 825
Copyright
Copyright © 2015 by the Applied Probability Trust 

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