Markov chain Monte Carlo with mixtures of singular distributions
Raphaël Gottardo, Adrian E. Raftery · 2004
Markov chain Monte Carlo (MCMC) methods for Bayesian computation are mostly used when the dominating measure is the Lebesgue measure, the counting measure or a product of these. Many Bayesian problems give rise to distributions that are not dominated by the Lebesgue measure or the counting measure alone. In this paper, we introduce a simple framework for using MCMC algorithms in Bayesian computation with mixtures of singular distributions. The idea is to find a common dominating measure that allows the use of traditional Metropolis-Hastings algorithms. We show how our formulation can be used in Bayesian model selection. Using our formulation, when the full conditionals are available, the Gibbs sampler can be used. We compare our formulation with the reversible jump approach, and show that the two are closely related. We give results for four examples, involving testing a normal mean, variable selection in regression, and hypothesis testing for differential expression under multiple conditions in gene expression data. This allows us to compare the three methods considered: Metropolis-Hastings with singular measures, Gibbs sampler with singular measures, and reversible jump. In our examples, we found the Gibbs sampler with singular measures to be more precise and to need considerably less computer time than the other methods.