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

TLDR

Computing a Bayes factor for testing H0: ψ=ψ0 with a nuisance parameter β requires careful prior specification, and automatic or reference prior methods based solely on model structure have been sought. This article applies a heuristic that sets the prior information on ψ equal to that of a single observation. After transforming β to be null‑orthogonal to ψ, the authors assign identical marginal priors to β under both hypotheses and use a normal prior for ψ. With a normal prior on ψ, the log Bayes factor is approximated by the Schwarz criterion with error O_p(n^−½), suggesting the criterion provides sensible approximations for nested hypotheses, while an elliptically Cauchy prior requires a constant correction, yielding a multidimensional generalization of Jeffreys’s method.

Abstract

Abstract To compute a Bayes factor for testing H 0: ψ = ψ0 in the presence of a nuisance parameter β, 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 ψ equal to the amount of information in a single observation. Then, after transforming β to be "null orthogonal" to ψ, we take the marginal priors on β to be equal under the null and alternative hypotheses. Doing so, and taking the prior on ψ 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 p (n −½), rather than the usual error of order O p (1). 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 ψ 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.