Stable Limit Laws for Marginal Probabilities from MCMC Streams: Acceleration of Convergence

In the Bayesian paradigm the marginal probability density function at the observed data vector is the key ingredient needed to compute Bayes factors and posterior probabilities of models and hypotheses. Although Markov chain Monte Carlo methods have simplified many calculations needed for the practical application of Bayesian methods, the problem of evaluating this marginal probability remains difficult. Newton and Raftery discovered that the harmonic mean of the likelihood function along an MCMC stream converges almostsurely but very slowly to the required marginal probability density. In this paper examples are shown to illustrate that these harmonic means converge in distribution to a one-sided stable law with index between one and two. Methods are proposed and illustrated for evaluating the required marginal probability density of the data from the limiting stable distribution, offering a dramatic acceleration in convergence over existing methods.