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February 2007 Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing
Yongtao Guan, Stephen M. Krone
Ann. Appl. Probab. 17(1): 284-304 (February 2007). DOI: 10.1214/105051606000000772


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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Yongtao Guan. Stephen M. Krone. "Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing." Ann. Appl. Probab. 17 (1) 284 - 304, February 2007.


Published: February 2007
First available in Project Euclid: 13 February 2007

zbMATH: 1139.65001
MathSciNet: MR2292588
Digital Object Identifier: 10.1214/105051606000000772

Primary: 65C05
Secondary: 65C40

Keywords: Cheeger’s inequality , Isoperimetric inequality , Markov chain , Metropolis-coupled MCMC , Monte Carlo , small world , spectral gap , state decomposition

Rights: Copyright © 2007 Institute of Mathematical Statistics


Vol.17 • No. 1 • February 2007
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