In a thought-provoking paper, Efron (2011) investigated the merit and limitation of an empirical Bayes method to correct selection bias based on Tweedie's formula first reported by \cite{Robbins:1956}. The exceptional virtue of Tweedie's formula for the normal distribution lies in its representation of selection bias as a simple function of the derivative of log marginal likelihood. Since the marginal likelihood and its derivative can be estimated from the data directly without invoking prior information, bias correction can be carried out conveniently. We propose a Bayesian hierarchical model for chi-squared data such that the resulting Tweedie's formula has the same virtue as that of the normal distribution. Because the family of noncentral chi-squared distributions, the common alternative distributions for chi-squared tests, does not constitute an exponential family, our results cannot be obtained by extending existing results. Furthermore, the corresponding Tweedie's formula manifests new phenomena quite different from those of the normal distribution and suggests new ways of analyzing chi-squared data.

翻译：Efron (2011) 在一份发人深省的文章中,Efron (2011年)调查了根据Tweedie首次报告的公式纠正选择偏差的经验型贝ys方法的优缺点和局限性。Tweedie对正常分布的公式的特殊优点在于它代表了选择偏差,这是日志边际可能性衍生物的一个简单函数。由于边际可能性及其衍生物可以直接从数据中估算,而不必援引先前的信息,因此可以方便地进行偏差纠正。我们提议了一种Bayesian等级模型,用于根据奇形数据进行分界。我们提议了一种Bayesian等级模型,使由此产生的Tweedie的公式与正常分布的公式具有相同的优点。由于非中性奇形色分布的家族,即彩色测试的共同替代分配方式不构成一个指数式家庭,因此我们无法通过扩大现有结果获得结果。此外,相应的Tweedie的公式显示了与正常分布模式截然不同的新现象,并提出了分析奇形数据的新方法。