On April 7th, the seminar group welcomed Dr. Florian Maire from the Université de Montréal. Florian introduced recent work detailing a method for comparing two or more competing MCMC algorithms. Details for Florian’s talk are below.
Title: Weak Peskun ordering for approximate MCMC comparison
Abstract: Despite the popularity of MCMC methods in Bayesian statistics and elsewhere, very few tools are available to establish a theoretical comparison between two (or more) competing MCMC algorithms. Often, comparison is carried out as a post-processing step after running each algorithm individually and by looking, for instance, at the integrated autocorrelation time or the expected squared jump distance. This is computationally costly and a rigorous interpretation of those criterions (and others) is missing. Instead, one would like (i) a criterion that gives a domination of one algorithm over another one in a well defined theoretical sense and (ii) knowing this criterion without simulating any Markov chain. The Peskun ordering (Peskun, 1973) is well known for achieving those two tasks. However, it is a restrictive criterion in that it is hard to show that one algorithm dominates another one, in the Peskun ordering sense. In this work, we propose a weaker version of the Peskun ordering which is more widely applicable and easier to establish, and this, at the price of giving only approximate comparison results. This weak Peskun ordering is applied to (partially) answer two recent questions in the MCMC literature, namely when does a non-reversible Metropolis random walk dominate a reversible one and when does a locally weighted Gibbs sampler dominate the random scan Gibbs sampler.