Bayes factors for composite hypotheses have difficulty in encoding vague prior knowledge, as improper priors cannot be used and objective priors may be subjectively unreasonable. To address these issues we revisit the posterior Bayes factor, in which the posterior distribution from the data at hand is re-used in the Bayes factor for the same data. We argue that this is biased when calibrated against proper Bayes factors, but propose adjustments to allow interpretation on the same scale. In the important case of a regular normal model, the bias in log scale is half the number of parameters. The resulting empirical Bayes factor is closely related to the widely applicable information criterion. We develop test-based empirical Bayes factors for several standard tests and propose an extension to multiple testing closely related to the optimal discovery procedure. For non-parametric tests the empirical Bayes factor is approximately 10 times the P-value. We propose interpreting the strength of Bayes factors on a logarithmic scale with base 3.73, reflecting the sharpest distinction between weaker and stronger belief. This provides an objective framework for interpreting statistical evidence, realising a Bayesian/frequentist compromise.

翻译：复合假设的贝叶斯因子在编码模糊先验知识时存在困难，因为不能使用非正常先验，且客观先验可能主观上不合理。为解决这些问题，我们重新审视了后验贝叶斯因子，该方法将来自当前数据的后验分布重新用于同一数据的贝叶斯因子中。我们论证了当根据标准贝叶斯因子校准时，这种方法存在偏差，但提出了调整方法以允许在同一尺度上进行解释。在正则正态模型这一重要情形下，对数尺度的偏差是参数数量的一半。由此产生的经验贝叶斯因子与广泛适用信息准则密切相关。我们针对若干标准检验开发了基于检验的经验贝叶斯因子，并提出了与最优发现过程密切相关的多重检验扩展方法。对于非参数检验，经验贝叶斯因子约为P值的10倍。我们建议以底数3.73的对数尺度解释贝叶斯因子的强度，这反映了弱信念与强信念之间最显著的区别。这为解释统计证据提供了一个客观框架，实现了贝叶斯/频率学派的折衷。