90

2012

928-934

Journal of the American Statistical Association

PDF

To compute a Bayes factor for testing Ho: $ = $,, in the presence of a nuisance parameter @, priors under the null and alternative hypotheses must be chosen. As in Bayesian estimation, an important problem has been to define automatic, or “reference,” methods for determining priors based only on the structure of the model. In this article we apply the heuristic device of taking the amount of information in the prior on $ equal to the amount of information in a single observation. Then, after transforming @ to be “null orthogonal” to $, we take the marginal priors on @ to be equal under the null and alternative hypotheses. Doing so, and taking the prior on $ to be Normal, we find that the log of the Bayes factor may be approximated by the Schwarz criterion with an error of order O,(n-’”)r,a ther than the usual error of order O,( I ) . This result suggests the Schwarz criterion should provide sensible approximate solutions to Bayesian testing problems, at least when the hypotheses are nested. When instead the prior on $is elliptically Cauchy, a constant correction term must be added to the Schwarz criterion; the result then becomes a multidimensional generalization of Jeffreys’s method.

EA 102

1397/11/06

DAshagha

104

431