Open Access
October 2013 Comparison inequalities and fastest-mixing Markov chains
James Allen Fill, Jonas Kahn
Ann. Appl. Probab. 23(5): 1778-1816 (October 2013). DOI: 10.1214/12-AAP886

Abstract

We introduce a new partial order on the class of stochastically monotone Markov kernels having a given stationary distribution π on a given finite partially ordered state space X. When KL in this partial order we say that K and L satisfy a comparison inequality. We establish that if K1,,Kt and L1,,Lt are reversible and KsLs for s=1,,t, then K1KtL1Lt. In particular, in the time-homogeneous case we have KtLt for every t if K and L are reversible and KL, and using this we show that (for suitable common initial distributions) the Markov chain Y with kernel K mixes faster than the chain Z with kernel L, in the strong sense that at every time t the discrepancy—measured by total variation distance or separation or L2-distance—between the law of Yt and π is smaller than that between the law of Zt and π.

Using comparison inequalities together with specialized arguments to remove the stochastic monotonicity restriction, we answer a question of Persi Diaconis by showing that, among all symmetric birth-and-death kernels on the path X={0,,n}, the one (we call it the uniform chain) that produces fastest convergence from initial state 0 to the uniform distribution has transition probability 1/2 in each direction along each edge of the path, with holding probability 1/2 at each endpoint.

We also use comparison inequalities:

(i) to identify, when is a given log-concave distribution on the path, the fastest-mixing stochastically monotone birth-and-death chain started at , and

(ii) to recover and extend a result of Peres and Winkler that extra updates do not delay mixing for monotone spin systems.

Among the fastest-mixing chains in (i), we show that the chain for uniform is slowest in the sense of maximizing separation at every time.

Citation

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James Allen Fill. Jonas Kahn. "Comparison inequalities and fastest-mixing Markov chains." Ann. Appl. Probab. 23 (5) 1778 - 1816, October 2013. https://doi.org/10.1214/12-AAP886

Information

Published: October 2013
First available in Project Euclid: 28 August 2013

zbMATH: 1288.60089
MathSciNet: MR3114917
Digital Object Identifier: 10.1214/12-AAP886

Subjects:
Primary: 60J10

Keywords: birth-and-death chains , comparison inequalities , fastest mixing , ladder game , log-concave distributions , Markov chains , Stochastic monotonicity

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.23 • No. 5 • October 2013
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