Simple Conditions for Metastability of Continuous Markov Chains

TL;DR

This paper provides simple, verifiable conditions under which continuous Markov chains exhibit metastability, offering precise spectral gap formulas and applying to practical algorithms like Metropolis-Hastings.

Contribution

It introduces easy-to-verify conditions for metastability in Markov chains and derives asymptotically exact spectral gap formulas for a class of examples.

Findings

Conditions verified for Metropolis-Hastings chains targeting mixture distributions

Provides asymptotically exact spectral gap formulas

Facilitates comparison of mixing times for different algorithms

Abstract

A family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit $\textit{metastable mixing}$ with $\textit{modes}$ $S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other mixing property) is very close to the worst conductance $\min(\Phi_{\beta}(S_{\beta}^{(1)}), \ldots, \Phi_{\beta}(S_{\beta}^{(k)}))$ of its modes. 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. Our work differs from existing work on metastability in that, for the class of examples we are interested in, it gives an asymptotically exact formula for the spectral gap (rather than a bound that can be very far from sharp) while at the same time giving technical conditions that are easier to verify for many statistical examples. Our bounds from this paper are used in a companion paper to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a random walk algorithm for multimodal target distributions.