Convergence of Stratified MCMC Sampling of Non-Reversible Dynamics

TL;DR

This paper introduces a stratified MCMC algorithm designed for non-reversible stochastic dynamics, proving its convergence and analyzing its fixed points and mixing properties, with potential applications in sampling complex systems.

Contribution

It generalizes existing milestoning and NEUS methods, providing convergence proofs and linking algorithm speed to process mixing rates and eigenvalue problems.

Findings

Proves convergence of the stratified MCMC method.

Shows the fixed point corresponds to the invariant measure.

Relates algorithm speed to mixing rates and eigenvalues.

Abstract

We present a form of stratified MCMC algorithm built with non-reversible stochastic dynamics in mind. It can also be viewed as a generalization of the exact milestoning method, or form of NEUS. We prove convergence of the method under certain assumptions, with expressions for the convergence rate in terms of the process's behavior within each stratum and large scale behavior between strata. We show that the algorithm has a unique fixed point which corresponds to the invariant measure of the process without stratification. We will show how the speeds of two versions of the new algorithm, one with an extra eigenvalue problem step and one without, relate to the mixing rate of a discrete process on the strata, and the mixing probability of the process being sampled within each stratum. The eigenvalue problem version also relates to local and global perturbation results of discrete Markov chains, such as those given by Van Koten, Weare et. al.