Statistica Sinica 22 (2012), 1233-1251

$\alpha$-STABLE LIMIT LAWS FOR HARMONIC MEAN

ESTIMATORS OF MARGINAL LIKELIHOODS

Robert L. Wolpert and Scott C. Schmidler

Duke University

Abstract: The task of calculating marginal likelihoods arises in a wide array of statistical inference problems, including the evaluation of Bayes factors for model selection and hypothesis testing. Although Markov chain Monte Carlo methods have simplified many posterior calculations needed for practical Bayesian analysis, the evaluation of marginal likelihoods remains difficult. We consider the behavior of the well-known harmonic mean estimator (Newton and Raftery (1994)) of the marginal likelihood, which converges almost-surely but may have infinite variance and so may not obey a central limit theorem.

We illustrate the convergence in distribution of the harmonic mean estimator in typical applications to a one-sided stable law with characteristic exponent $1 < \alpha < 2$. While the harmonic mean estimator does converge almost surely, we show that it does so at rate $n^{-\epsilon}$ where $\epsilon=(\alpha-1) /\alpha$ is often as small as $0.10$ or $0.01$. In such a case, the reduction of Monte Carlo sampling error by a factor of two requires increasing the Monte Carlo sample size by a factor of $2^{1/\epsilon}$, or in excess of $2.5\cdot 10^{30}$ when $\epsilon=0.01$, rendering the method entirely untenable. We explore the possibility of estimating the parameters of the limiting stable distribution to provide accelerated convergence.

Key words and phrases: Alpha stable, Bayes factors, bridge sampling, harmonic mean, marginal likelihood, model averaging.