We explicitly define the notion of (exact or approximate) compound e-values which have been implicitly presented and extensively used in the recent multiple testing literature. We show that every FDR controlling procedure can be recovered by instantiating the e-BH procedure with certain compound e-values. Since compound e-values are closed under averaging, this allows for combination and derandomization of any FDR procedure. We then point out and exploit the connections to empirical Bayes. In particular, we use the fundamental theorem of compound decision theory to derive the log-optimal simple separable compound e-value for testing a set of point nulls against point alternatives: it is a ratio of mixture likelihoods. We extend universal inference to the compound setting. As one example, we construct approximate compound e-values for multiple t-tests, where the (nuisance) variances may be different across hypotheses. We provide connections to related notions in the literature stated in terms of p-values.

翻译：我们明确界定了（精确或近似）复合e值的概念，该概念在近期多重检验文献中已被隐式提出并广泛应用。我们证明，所有FDR控制方法均可通过将e-BH方法实例化为特定复合e值而得以重构。由于复合e值在平均运算下具有封闭性，这使得任意FDR方法均可进行组合与去随机化处理。随后我们指出并利用其与经验贝叶斯的关联性，特别通过运用复合决策理论基本定理，推导出针对点零假设与点备择假设检验的对数最优简单可分离复合e值：其本质为混合似然比。我们将通用推断扩展至复合设定，并以多重t检验为例构建近似复合e值（其中不同假设间的（冗余）方差可存在差异）。最后，我们建立了与文献中基于p值相关概念的对应关系。