A Reference Bayesian Test for Nested Hypotheses
And its Relationship to the Schwarz Criterion

Robert E. Kass and Larry Wasserman


To compute a Bayes factor for testing 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 paper 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 , rather than the usual error of order . 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.

Keywords: Bayes Information Criterion, Laplace's method, Model selection, Null-orthogonal parameters, Orthogonal parameters.

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