The Annals of Applied Probability

Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing

Yongtao Guan and Stephen M. Krone

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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.

Article information

Ann. Appl. Probab., Volume 17, Number 1 (2007), 284-304.

First available in Project Euclid: 13 February 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods
Secondary: 65C40: Computational Markov chains

Markov chain Monte Carlo small world spectral gap Cheeger’s inequality state decomposition isoperimetric inequality Metropolis-coupled MCMC


Guan, Yongtao; Krone, Stephen M. Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing. Ann. Appl. Probab. 17 (2007), no. 1, 284--304. doi:10.1214/105051606000000772.

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