Circularly-Coupled Markov Chain Sampling

Abstract

I show how to run an N-time-step Markov chain simulation in a circular fashion, so that the state at time 0 follows the state at time N-1 in the same way as states at times t follow those at times t-1 for 0<t<N. This wrap-around of the chain is achieved using a coupling procedure, and produces states that all have close to the equilibrium distribution of the Markov chain, under the assumption that coupled chains are likely to coalesce in less than N/2 iterations. This procedure therefore automatically eliminates the initial portion of the chain that would otherwise need to be discarded to get good estimates of equilibrium averages. The assumption of rapid coalescence can be tested using auxiliary chains started at times spaced between 0 and N. When multiple processors are available, such auxiliary chains can be simulated in parallel, and pieced together to give the circularly-coupled chain, in less time than a sequential simulation would have taken, provided that coalescence is indeed rapid. The practical utility of these procedures is dependent on the development of good coupling schemes. I show how a specialized random-grid Metropolis algorithm can be used to produce the required exact coalescence. On its own, this method is not efficient in high dimensions, but it can be used to produce exact coalescence once other methods have brought the coupled chains close together. I investigate how well this combined scheme works with standard Metropolis, Langevin, and Gibbs sampling updates. Using such strategies, I show that circular coupling can work effectively in a Bayesian logistic regression problem.