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Simple conditions for metastability of continuous Markov chains

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  25 February 2021

Oren Mangoubi ,
Natesh Pillai  and
Aaron Smith
Oren Mangoubi*
Affiliation:
Worcester Polytechnic Institute
Natesh Pillai*
Affiliation:
Harvard University
Aaron Smith*
Affiliation:
University of Ottawa
*
*Postal address: Worcester Polytechnic Institute, 100 Institute Road, Worcester, Massachusetts, USA. Email address: [email protected]
**Postal address: Department of Statistics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA.
***Postal address: Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa ON K1N 7N5, Canada.
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Abstract

A family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit metastable mixing with modes$S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other mixing property) is very close to the worst conductance $\min\!\big(\Phi_{\beta}\big(S_{\beta}^{(1)}\big), \ldots, \Phi_{\beta}\big(S_{\beta}^{(k)}\big)\big)$ of its modes for all large values of $\beta$. We give simple sufficient conditions for a family of Markov chains to exhibit metastability in this sense, and verify that these conditions hold for a prototypical Metropolis–Hastings chain targeting a mixture distribution. The existing metastability literature is large, and our present work is aimed at filling the following small gap: finding sufficient conditions for metastability that are easy to verify for typical examples from statistics using well-studied methods, while at the same time giving an asymptotically exact formula for the spectral gap (rather than a bound that can be very far from sharp). Our bounds from this paper are used in a companion paper (O. Mangoubi, N. S. Pillai, and A. Smith, arXiv:1808.03230) to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a random walk algorithm for multimodal target distributions.

Keywords

MetastabilityMarkov chain Monte Carlo (MCMC)spectral gapmultimodal distribution

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 65C40: Computational Markov chains
Type
Research Papers
Information
Journal of Applied Probability , Volume 58 , Issue 1 , March 2021 , pp. 83 - 105
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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