Gönen Mithat, Johnson Wesley O, Lu Yonggang, Westfall Peter H
Memorial Sloan-Kettering Cancer Center.
Department of Statistics, UC Irvine.
Am Stat. 2019;73(1):22-31. doi: 10.1080/00031305.2017.1322142. Epub 2018 May 10.
Many Bayes factors have been proposed for comparing population means in two-sample (independent samples) studies. Recently, Wang and Liu (2015) presented an "objective" Bayes factor (BF) as an alternative to a "subjective" one presented by Gönen et al. (2005). Their report was evidently intended to show the superiority of their BF based on "undesirable behavior" of the latter. A wonderful aspect of Bayesian models is that they provide an opportunity to "lay all cards on the table." What distinguishes the various BFs in the two-sample problem is the choice of priors (cards) for the model parameters. This article discusses desiderata of BFs that have been proposed, and proposes a new criterion to compare BFs, no matter whether subjectively or objectively determined: A BF may be preferred if it correctly classifies the data as coming from the correct model most often. The criterion is based on a famous result in classification theory to minimize the total probability of misclassification. This criterion is objective, easily verified by simulation, shows clearly the effects (positive or negative) of assuming particular priors, provides new insights into the appropriateness of BFs in general, and provides a new answer to the question, "Which BF is best?"
在双样本（独立样本）研究中，已经提出了许多用于比较总体均值的贝叶斯因子。最近，王和刘（2015年）提出了一种“客观”贝叶斯因子（BF），作为戈嫩等人（2005年）提出的“主观”贝叶斯因子的替代方案。他们的报告显然旨在基于后者的“不良行为”来展示其BF的优越性。贝叶斯模型的一个美妙之处在于它们提供了一个“摊牌”的机会。在双样本问题中，各种BF的区别在于模型参数的先验（牌）选择。本文讨论了已提出的BF的 desiderata，并提出了一种比较BF的新标准，无论其是主观还是客观确定的：如果一个BF最常将数据正确分类为来自正确模型，则可能更受青睐。该标准基于分类理论中的一个著名结果，以最小化误分类的总概率。该标准是客观的，通过模拟很容易验证，清楚地显示了假设特定先验的影响（正面或负面），总体上为BF的适用性提供了新的见解，并为“哪个BF最好？”这个问题提供了新的答案。
