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.
Publié le : 2007-02-14
Classification:
Markov chain,
Monte Carlo,
small world,
spectral gap,
Cheeger’s inequality,
state decomposition,
isoperimetric inequality,
Metropolis-coupled MCMC,
65C05,
65C40
@article{1171377185,
author = {Guan, Yongtao and Krone, Stephen M.},
title = {Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing},
journal = {Ann. Appl. Probab.},
volume = {17},
number = {1},
year = {2007},
pages = { 284-304},
language = {en},
url = {http://dml.mathdoc.fr/item/1171377185}
}
Guan, Yongtao; Krone, Stephen M. Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing. Ann. Appl. Probab., Tome 17 (2007) no. 1, pp. 284-304. http://gdmltest.u-ga.fr/item/1171377185/