Simple conditions for metastability of continuous Markov chains. (March 2021)

• Record Type: Journal Article
• Authors: Mangoubi, Oren; Pillai, Natesh; Smith, Aaron
• Is Part Of: Journal of applied probability. Volume 58:Number 1(2021)
• Page Start: 83
• Page End: 105
• Publication Date: 2021-03
• Subjects: Metastability, -- Markov chain Monte Carlo (MCMC), -- spectral gap, -- multimodal distribution; 60J05, -- 65C40; 519.2
• DOI: 10.1017/jpr.2020.83
• Languages: English
• ISSNs: 0021-9002

Abstract

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.