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Suppressing random walks in Markov chain Monte Carlo using ordered overrelaxation.pdf

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Suppressing random walks in Markov chain Monte Carlo using ordered overrelaxation

Technical Report No. 9508, Department of Statistics, University of TorontoSuppressing Random Walks in Markov ChainMonte Carlo Using Ordered OverrelaxationRadford M. NealDept. of Statistics and Dept. of Computer ScienceUniversity of TorontoToronto, Ontario, CanadaWorld Wide Web: /radfordE-mail: radford@21 June 1995Markov chain Monte Carlo methods such as Gibbs sampling and simple forms of theMetropolis algorithm typically move about the distribution being sampled via a ran-dom walk. For the complex, high-dimensional distributions commonly encountered inBayesian inference and statistical physics, the distance moved in each iteration of thesealgorithms will usually be small, because it is dicult or impossible to transform theproblem to eliminate dependencies between variables. The ineciency inherent in takingsuch 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 n2 itera-tions. Such random walks can sometimes be suppressed using \overrelaxed variants ofGibbs sampling (a.k.a. the heatbath algorithm), but such methods have hitherto beenlargely restricted to problems where all the full conditional distributions are Gaussian.I present an overrelaxed Markov chain Monte Carlo algorithm based on order statisticsthat is more widely applicable. In particular, the algorithm can be applied whenever thefull conditional distributions are such that their cumulative distribution functions andinverse cumulative distribution functions can be eciently computed. The method isdemonstrated on an inference problem for a simple hierarchical Bayesian model. 1 1 IntroductionMarkov chain Monte Carlo methods are used to estimate the expectations of variousfunctions of a state, x = (x1; . . . ; xN), with respect to a distribution given by somedensity function, (x). Typically, the dimensionality, N , is large, and the density (x) isof a complex form, in which the components of x are hig