Mixing and hitting times for Gibbs samplers and other non-Feller processes

Abstract

The hitting and mixing times are two often-studied quantities associated with Markov chains. Yuval Peres, Perla Sousi and Roberto Oliveira showed that the mixing times and “worst-case” hitting times of reversible Markov chains on finite state spaces are “equivalent”—that is, equal up to some universal multiplicative constant. We have extended this strong connection between mixing and hitting times to Markov chains satisfying the strong Feller property in an earlier work. In the present paper, we further extend the results to include Metropolis–Hastings chains, the popular Gibbs sampler (from statistics), and Glauber dynamics (from statistical physics), which make “one-dimensional” updates and thus do not satisfy the strong Feller property. We also apply this result to obtain decomposition bounds for such Markov chains. Our main tools come from nonstandard analysis.