Ensemble MCMC

I’m glad to have managed, before teaching starts again, to have finished a Technical Report (available here or at arxiv.org) with what may be my most unwieldy title ever:

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

I wanted the title to mention all three of the nested ideas in the paper.  Actually, I wasn’t able to fit in a fourth, most general, idea, of MCMC methods based on caching and mapping (see here).   Here is the abstract:

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 in {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.

I’ve also posted the programs used to produce the results.  These haven’t been tested much beyond their use for the paper, but I hope to incorporate them into a general MCMC package in R (also including programs accompanying my review of Hamiltonian Monte Carlo).   That’s my next project to do in whatever time I have available after teaching, administration, and a three-year-old daughter, along with more efforts to speed up R, so the that this MCMC package won’t be too slow.