## The Annals of Applied Probability

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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 13 February 2007

https://projecteuclid.org/euclid.aoap/1171377185

Digital Object Identifier
doi:10.1214/105051606000000772

Mathematical Reviews number (MathSciNet)
MR2292588

Zentralblatt MATH identifier
1139.65001

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

#### Citation

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. https://projecteuclid.org/euclid.aoap/1171377185

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