Gibbs sampling is a popular Markov chain Monte Carlo method that samples from a distribution on n state variables by repeatedly sampling from the conditional distribution of one variable, xi, given the other variables, x-i, either choosing i randomly, or updating sequentially using some systematic or random order for i. When xi is discrete, a Gibbs sampling update may choose a new value that is the same as the old value. A theorem of Peskun indicates that, when i is chosen randomly, a reversible method that reduces the probability of such self transitions, while increasing the probabilities of transitioning to each of the other values, will decrease the asymptotic variance of estimates of expectations of functions of the state. This has inspired two modified Gibbs sampling methods, originally due to Frigessi, Hwang, and Younes and to Liu, though these do not always reduce self transitions to the minimum possible. Methods that do reduce the probability of self transitions to the minimum, but do not satisfy the conditions of Peskun's theorem, have also been devised, by Suwa and Todo, some of which are reversible and some not. I review and relate these past methods, and introduce a broader class of reversible methods, including that of Frigessi, \textit{et al.}, based on what I call ``antithetic modification'', which also reduce asymptotic variance compared to Gibbs sampling, even when not satisfying the conditions of Peskun's theorem. A modification of one method in this class, which I denote as ZDNAM, reduces self transitions to the minimum possible, while still always reducing asymptotic variance compared to Gibbs sampling. I introduce another new class of non-reversible methods based on slice sampling that can also minimize self transition probabilities. I provide explicit, efficient implementations of all these methods, and compare the performance of Gibbs sampling and these modified Gibbs sampling methods in simulations of a 2D Potts model, a Bayesian mixture model, and a belief network with unobserved variables. The assessments look at both random selection of i, and several sequential update schemes. Sequential updates using methods that minimize self transition probabilities are found to usually be superior, with ZDNAM often performing best. There is evidence that the non-reversibility produced by sequential updating can be beneficial, but no consistent benefit is seen from the individual updates being done by a non-reversible method.
Technical report, 26 March 2024, 84 pages: pdf.
Also available at arXiv.org.
Neal, R. M. and Rosenthal, J. S. (2023) ``Efficiency of reversible MCMC methods: Elementary derivations and applications to composite methods'', Technical Report, 24 pages: abstract, pdf.