MCMC Using Ensembles of States for Problems with Fast and Slow Variables such as Gaussian Process Regression

I introduce a Markov chain Monte Carlo (MCMC) scheme in which sampling from a distribution with density π(x) is done using updates operating on an “ensemble” of states. The current state x is first stochastically mapped to an ensemble, (x(1), . . . , x(K)). This ensemble is then updated using MCMC updates that leave invariant a suitable ensemble density, ρ(x(1), . . . , x(K)), defined in terms of π(x(i)) for i = 1, . . . ,K. Finally a single state is stochastically selected from the ensemble after these updates. Such ensemble MCMC updates can be useful when characteristics of π and the ensemble permit π(x(i)) for all i ∈ {1, . . . ,K}, to be computed in less than K times the amount of computation time needed to compute π(x) for a single x. One common situation of this type is when changes to some “fast” variables allow for quick re-computation of the density, whereas changes to other “slow” variables do not. Gaussian process regression models are an example of this sort of problem, with an overall scaling factor for covariances and the noise variance being fast variables. I show that ensemble MCMC for Gaussian process regression models can indeed substantially improve sampling performance. Finally, I discuss other possible applications of ensemble MCMC, and its relationship to the “multiple-try Metropolis” method of Liu, Liang, and Wong and the “multiset sampler” of Leman, Chen, and Lavine. Introduction In this paper, I introduce a class of Markov chain Monte Carlo methods that utilize a state space that is the K-fold Cartesian product of the space of interest — ie, although our interest is in sampling for x in the space X , we use MCMC updates that operate on (x(1), . . . , x(K)) in the space Y = X . Several such methods have previously been proposed — for example, Adaptive Directive Sampling