Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing
Yongtao Guan and Stephen M. Krone
Source: Ann. Appl. Probab. Volume 17, Number 1
(2007), 284-304.
Abstract
We compare convergence rates of Metropolis–Hastings chains to multi-modal target distributions when the proposal distributions can be of “local” and “small world” type. In particular, we show that by adding occasional long-range jumps to a given local proposal distribution, one can turn a chain that is “slowly mixing” (in the complexity of the problem) into a chain that is “rapidly mixing.” To do this, we obtain spectral gap estimates via a new state decomposition theorem and apply an isoperimetric inequality for log-concave probability measures. We discuss potential applicability of our result to Metropolis-coupled Markov chain Monte Carlo schemes.
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Keywords: Markov chain; Monte Carlo; small world; spectral gap; Cheeger’s inequality; state decomposition; isoperimetric inequality; Metropolis-coupled MCMC
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1171377185
Digital Object Identifier: doi:10.1214/105051606000000772
Mathematical Reviews number (MathSciNet): MR2292588
Zentralblatt MATH identifier: 1139.65001
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