Coupling and Decoupling to bound an approximating Markov Chain

Abstract

This simple note lays out a few observations which are well known in many ways but may not have been said in quite this way before. The basic idea is that when comparing two different Markov chains it is useful to couple them is such a way that they agree as often as possible. We construct such a coupling and analyze it by a simple dominating chain which registers if the two processes agree or disagree. We find that this imagery is useful when thinking about such problems. We are particularly interested in comparing the invariant measures and long time averages of the processes. However, since the paths agree for long runs, it also provides estimates on various stopping times such as hitting or exit times. We also show that certain bounds are tight. Finally, we provide a simple application to a Markov Chain Monte Carlo algorithm and show numerically that the results of the paper show a good level of approximation at considerable speed up by using an approximating chain rather than the original sampling chain.