Abstract

In a thought-provoking paper, Efron investigated the merit and limitation of an empirical Bayes method to correct selection bias based on Tweedie’s formula first reported in the study by Robbins. 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.

摘要

在一篇发人深省的论文中，Efron 根据 Robbins 的研究中首次报道的 Tweedie 公式调查了经验贝叶斯方法在纠正选择偏差方面的优点和局限性。Tweedie 正态分布公式的特殊优点在于它将选择偏差表示为对数边际似然导数的简单函数。由于可以直接从数据中估计边际似然及其导数，而无需调用先验信息，因此可以方便地进行偏差校正。我们为卡方数据提出了贝叶斯层次模型，使得生成的 Tweedie 公式具有与正态分布相同的优点。因为非中心卡方分布族，卡方检验的常见替代分布，不构成指数族，我们的结果不能通过扩展现有结果来获得。此外，相应的特威迪公式表现出与正态分布截然不同的新现象，并提出了分析卡方数据的新方法。