Sharp entrywise perturbation bounds for Markov chains

TL;DR

This paper develops sharp, entrywise perturbation bounds for Markov chains' invariant distributions, revealing differential sensitivities to transition matrix perturbations without structural assumptions.

Contribution

It introduces novel, sharp bounds on the relative error of invariant distributions that are easy to compute and interpret, applicable to general Markov chains.

Findings

Bounds are sharp and do not require structural assumptions.

Sensitivity varies significantly across different entries of the transition matrix.

Bounds can be interpreted through hitting times for intuitive understanding.

Abstract

For many Markov chains of practical interest, the invariant distribution is extremely sensitive to perturbations of some entries of the transition matrix, but insensitive to others; we give an example of such a chain, motivated by a problem in computational statistical physics. We have derived perturbation bounds on the relative error of the invariant distribution that reveal these variations in sensitivity. Our bounds are sharp, we do not impose any structural assumptions on the transition matrix or on the perturbation, and computing the bounds has the same complexity as computing the invariant distribution or computing other bounds in the literature. Moreover, our bounds have a simple interpretation in terms of hitting times, which can be used to draw intuitive but rigorous conclusions about the sensitivity of a chain to various types of perturbations.