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^{-ε} 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
2^{1/ε}, or in excess of 2.5× 10^{30} 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,
}