Suppressing Random Walks in Markov Chain Monte Carlo Using Ordered Overrelaxation

TL;DR

This paper introduces a new overrelaxed MCMC algorithm based on order statistics that reduces random walk behavior, improving sampling efficiency for complex, high-dimensional distributions beyond Gaussian cases.

Contribution

A novel overrelaxed MCMC method using order statistics that applies to a broader class of distributions with computable CDFs and inverse CDFs, extending beyond Gaussian cases.

Findings

Reduces random walk inefficiency in MCMC sampling.

Applicable to distributions with computable CDFs and inverse CDFs.

Demonstrated on hierarchical Bayesian model inference.

Abstract

Markov chain Monte Carlo methods such as Gibbs sampling and simple forms of the Metropolis algorithm typically move about the distribution being sampled via a random walk. For the complex, high-dimensional distributions commonly encountered in Bayesian inference and statistical physics, the distance moved in each iteration of these algorithms will usually be small, because it is difficult or impossible to transform the problem to eliminate dependencies between variables. The inefficiency inherent in taking such small steps is greatly exacerbated when the algorithm operates via a random walk, as in such a case moving to a point n steps away will typically take around n^2 iterations. Such random walks can sometimes be suppressed using ``overrelaxed'' variants of Gibbs sampling (a.k.a. the heatbath algorithm), but such methods have hitherto been largely restricted to problems where all the full conditional distributions are Gaussian. I present an overrelaxed Markov chain Monte Carlo algorithm based on order statistics that is more widely applicable. In particular, the algorithm can be applied whenever the full conditional distributions are such that their cumulative distribution functions and inverse cumulative distribution functions can be efficiently computed. The method is demonstrated on an inference problem for a simple hierarchical Bayesian model.