Approximate Bayes Factors When a Mode Occurs on the Boundary

Bayes factors, measuring the strength of evidence in favor of the null, are often used in the Bayesian approach to testing hypotheses. Laplace approximations to Bayes factors are convenient and quite accurate in many contexts. However, one usual assumption—the existence of an interior mode—does not always hold. The posterior mode can occur at the boundary of the parameter space. This article discusses the boundary mode Laplace's method for boundary modes, uses it to approximate the Bayes factor, and presents a modification to the Schwarz criterion. The sensitivity of Bayes factor to the choice of prior on the nuisance parameter in testing nested models is also investigated. Results are illustrated for the case of testing extrabinomial variability.