α-Stable Limit Laws for Harmonic Mean Estimators of Marginal Likelihoods

α-Stable Limit Laws for
Harmonic Mean Estimators
of Marginal Likelihoods

Robert L. Wolpert and Scott C. Schmidler

Duke University Department of Statistical Science

Rev: Jun 2011

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 of the marginal likelihood (Newton and Raftery, 1994), which converges almost-surely but may have infinite variance and so may not obey a central limit theorem. We give examples illustrating the convergence in distribution of the harmonic mean estimator to a one-sided stable law with characteristic exponent 1<α<2. While the harmonic mean estimator does converge almost surely, we show that it does so at rate n where ε=1-α-1 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 21/ε, or in excess of 2.5× 1030 when ε=0.01. We explore the possibility of estimating the parameters of the limiting stable distribution to provide accelerated convergence.

Keywords: α-stable, Bayes factors, bridge sampling, harmonic mean, marginal likelihood, model averaging.


The manuscript is available in PDF format (331kb).


Cite as:

@Article{Wolp:Schm:2012,
      Author = "Robert L. Wolpert and Scott C. Schmidler",
       Title = "$\alpha$-Stable Limit Laws for Harmonic Mean Estimators
                of Marginal Likelihoods",
     Journal = "Statistica Sinica",
      Volume = 22,
      Number = 3,
        Note = "In press; preprint at \url
                {http://ftp.stat.duke.edu/WorkingPapers/10-19.html}",
         DOI = "10.5705/ss.2010.221",
        Year = 2012,
}