A Reference Bayesian Test for Nested Hypotheses and its Relationship to the Schwarz Criterion
提出一种自动确定先验分布的参考方法，用于嵌套假设的贝叶斯因子计算，并证明其对数贝叶斯因子可由施瓦茨准则近似，误差阶为O_p(n^{-1/2})。
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.