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Weak Convergence Rates of Population Versus Single-Chain Stochastic Approximation MCMC Algorithms

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

Published online by Cambridge University Press:  22 February 2016

Qifan Song ,
Mingqi Wu  and
Faming Liang
Qifan Song*
Affiliation:
Texas A&M University
Mingqi Wu*
Affiliation:
Shell Global Solutions (US) Inc.
Faming Liang*
Affiliation:
Texas A&M University
*
∗ Current address: Department of Statistics, Purdue University, West Lafayette, IN 47907, USA.
∗∗ Postal address: Shell Technology Center Houston, 3333 Highway 6 South, Houston, TX 77082, USA.
∗∗∗ Current address: Department of Biostatistics, University of Florida, Gainesville, FL 321611, USA. Email address: [email protected]
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Abstract

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In this paper we establish the theory of weak convergence (toward a normal distribution) for both single-chain and population stochastic approximation Markov chain Monte Carlo (MCMC) algorithms (SAMCMC algorithms). Based on the theory, we give an explicit ratio of convergence rates for the population SAMCMC algorithm and the single-chain SAMCMC algorithm. Our results provide a theoretic guarantee that the population SAMCMC algorithms are asymptotically more efficient than the single-chain SAMCMC algorithms when the gain factor sequence decreases slower than O(1 / t), where t indexes the number of iterations. This is of interest for practical applications.

Keywords

Asymptotic normalityMarkov chain Monte Carlostochastic approximationMetropolis-Hastings algorithm

MSC classification

Primary: 60J22: Computational methods in Markov chains
Secondary: 65C05: Monte Carlo methods
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 1059 - 1083
Copyright
© Applied Probability Trust 

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