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Conductance bounds on the L2 convergence rate of Metropolis algorithms on unbounded state spaces

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Søren F. Jarner  and
Wai Kong Yuen
Søren F. Jarner*
Affiliation:
The Danish Labour Market Supplementary Pension
Wai Kong Yuen*
Affiliation:
Brock University
*
∗ Postal address: Quantitative Research, Fixed Income, The Danish Labour Market Supplementary Pension, Kongens Vaenge 8, DK-3400 Hilleroed, Denmark. Email address: sj@atp.dk
∗∗ Postal address: Department of Mathematics, Brock University, St Catharines, Ontario, Canada L2S 3A1. Email address: jyuen@brocku.ca
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Abstract

In this paper we derive bounds on the conductance and hence on the spectral gap of a Metropolis algorithm with a monotone, log-concave target density on an interval of ℝ. We show that the minimal conductance set has measure ½ and we use this characterization to bound the conductance in terms of the conductance of the algorithm restricted to a smaller domain. Whereas previous work on conductance has resulted in good bounds for Markov chains on bounded domains, this is the first conductance bound applicable to unbounded domains. We then show how this result can be combined with the state-decomposition theorem of Madras and Randall (2002) to bound the spectral gap of Metropolis algorithms with target distributions with monotone, log-concave tails on ℝ.

Keywords

ConductanceMarkov chain Monte CarloMetropolis algorithmrates of convergencespectral gap

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 243 - 266
Copyright
Copyright © Applied Probability Trust 2004 

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