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 I 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. I 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. I develop test-based empirical Bayes factors for several standard tests and propose an extension to multiple testing closely related to the optimal discovery procedure. When only a P-value is available, an approximate empirical Bayes factor is 10p. I 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, and realises a Bayesian/frequentist compromise.

翻译：复合假设的贝叶斯因子难以编码模糊的先验知识，因为不适当的先验分布无法使用，而客观先验可能主观上不合理。为解决这些问题，我重新审视了后验贝叶斯因子——即利用当前数据的后验分布重新计算同一数据的贝叶斯因子。我论证了该方法在针对适当贝叶斯因子校准时存在偏差，但提出了调整方案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，对数尺度上的偏差为参数数量的一半。由此得到的经验贝叶斯因子与广泛适用信息准则密切相关。我为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程序密切相关的多重检验扩展方法。当仅可获取P值时，近似经验贝叶斯因子为10p。我建议以3.73为底的对数尺度解释贝叶斯因子的强度，这反映了较弱信念与较强信念之间的最显著区分。这为解释统计证据提供了客观框架，并实现了贝叶斯/频率学派方法的折衷。