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:综合假设的贝叶因因素难以将先前知识模糊起来,因为不能使用不当的前科,客观的前科可能主观上不合理。为了解决这些问题,我们重新审视后贝叶因因素,即巴伊斯系数对同一数据重新使用手头数据后部分布。我们认为,在根据适当的贝叶因因素校准时,这是有偏颇的,但建议作出调整,以便能够在同一尺度上进行解释。在正常正常模式的重要情况下,日志比例的偏差是参数数目的一半。由此得出的实证贝因因素与广泛适用的信息标准密切相关。我们为若干标准测试制定了基于试验的经验贝因因素,并提议扩展为与最佳发现程序密切相关的多次测试。非参数测试中,实证海湾系数大约是P值的10倍。我们提议用基数3.73来解释对等比例上的贝伊因因素的强度,以反映较弱和较强的信念之间的最鲜明区别。这为解释统计证据提供了客观的框架,实现贝伊/后方的妥协提供了一种客观框架。</s>