Perturbation theory for Markov chains via Wasserstein distance

TL;DR

This paper develops bounds on how small changes in Markov chain transitions affect their distributions, especially for Wasserstein ergodic chains, with applications to approximate MCMC methods in big data analysis.

Contribution

It introduces new bounds for Markov chain perturbations using Wasserstein distance, applicable to ergodic chains and approximate MCMC algorithms, under weak conditions.

Findings

Bounds are tight for autoregressive models.

Quantitative estimates are provided for approximate Metropolis-Hastings.

The approach applies Lyapunov functions to weakly ergodic chains.

Abstract

Perturbation theory for Markov chains addresses the question how small differences in the transitions of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the $n$th step distributions of two Markov chains when one of them satisfies a Wasserstein ergodicity condition. Our work is motivated by the recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the analysis of big data sets. By using an approach based on Lyapunov functions, we provide estimates for geometrically ergodic Markov chains under weak assumptions. In an autoregressive model, our bounds cannot be improved in general. We illustrate our theory by showing quantitative estimates for approximate versions of two prominent MCMC algorithms, the Metropolis-Hastings and stochastic Langevin algorithms.