Mixing Times and Hitting Times for General Markov Processes

TL;DR

This paper extends the known relationship between mixing and hitting times from finite to general state space Markov chains with the strong Feller property, using nonstandard analysis, and applies it to MCMC algorithms.

Contribution

It generalizes the equivalence of mixing and hitting times to broader Markov chains and demonstrates practical bounds for MCMC methods.

Findings

Extended the equivalence to general state spaces with strong Feller property.

Provided bounds on MCMC chain mixing times like Gibbs and Metropolis-Hastings.

Utilized nonstandard analysis techniques for the extension.

Abstract

The hitting and mixing times are two fundamental quantities associated with Markov chains. In Peres and Sousi[PS2015] and Oliveira[Oli2012], the authors show that the mixing times and "worst-case" hitting times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on general state spaces that satisfy the strong Feller property. Finally, we show that this asymptotic equivalence can be used to find bounds on the mixing times of a large class of Markov chains used in MCMC, such as typical Gibbs samplers and Metroplis-Hastings chains, even though they usually do not satisfy the strong Feller property.