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I introduce a Markov chain Monte Carlo (MCMC) scheme in which sampling from adistribution with density pi(x) is done using updates operating on an"ensemble" of states. The current state x is first stochastically mapped to anensemble, x^{(1)},...,x^{(K)}. This ensemble is then updated using MCMC updatesthat leave invariant a suitable ensemble density, rho(x^{(1)},...,x^{(K)}),defined in terms of pi(x^{(i)}) for i=1,...,K. Finally a single state isstochastically selected from the ensemble after these updates. Such ensembleMCMC updates can be useful when characteristics of pi and the ensemble permitpi(x^{(i)}) for all i in {1,...,K}, to be computed in less than K times theamount of computation time needed to compute pi(x) for a single x. One commonsituation of this type is when changes to some "fast" variables allow for quickre-computation of the density, whereas changes to other "slow" variables donot. Gaussian process regression models are an example of this sort of problem,with an overall scaling factor for covariances and the noise variance beingfast variables. I show that ensemble MCMC for Gaussian process regressionmodels can indeed substantially improve sampling performance. Finally, Idiscuss other possible applications of ensemble MCMC, and its relationship tothe "multiple-try Metropolis" method of Liu, Liang, and Wong and the "multisetsampler" of Leman, Chen, and Lavine.

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