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

TL;DR

This paper introduces an ensemble-based MCMC method that efficiently samples from distributions with fast and slow variables, demonstrated on Gaussian process regression models, improving sampling performance.

Contribution

The paper proposes a novel ensemble MCMC scheme that leverages fast variable updates, enhancing sampling efficiency for complex models like Gaussian processes.

Findings

Ensemble MCMC significantly improves sampling efficiency in Gaussian process regression.

The method exploits fast and slow variable structure to reduce computational cost.

Potential applications extend to other models with similar variable update characteristics.

Abstract

I introduce a Markov chain Monte Carlo (MCMC) scheme in which sampling from a distribution with density pi(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, rho(x^{(1)},...,x^{(K)}), defined in terms of pi(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 pi and the ensemble permit pi(x^{(i)}) for all i in {1,...,K}, to be computed in less than K times the amount of computation time needed to compute pi(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.