Empirical partially Bayes multiple testing and compound χ2 decisions
本文研究了高通量生物学中常用的经验部分贝叶斯多重检验方法，证明其条件p值满足特定公式，且用非参数最大似然估计先验时能近似达到参数速率，结合Benjamini-Hochberg过程可渐近控制错误发现率。
A common task in high-throughput biology is to screen for associations across thousands of units of interest, for example, genes or proteins. Often, the data for each unit are modeled as Gaussian measurements with unknown mean and variance and are summarized as per-unit sample averages and sample variances. The downstream goal is multiple testing for the means. In this domain, it is routine to “moderate” (i.e., to shrink) the sample variances through parametric empirical Bayes methods before computing p-values for the means. Such an approach is asymmetric in that a prior is posited and estimated for the nuisance parameters (variances) but not the primary parameters (means). Our work initiates the formal study of this paradigm, which we term “empirical partially Bayes multiple testing.” In this framework, if the prior for the variances were known, one could proceed by computing p-values conditional on the sample variances—a strategy called partially Bayes inference by Sir David Cox. We show that these conditional p-values satisfy an Eddington/Tweedie-type formula and are approximated at nearly-parametric rates when the prior is estimated by nonparametric maximum likelihood. The estimated p-values can be used with the Benjamini–Hochberg procedure to guarantee asymptotic control of the false discovery rate. Even in the compound setting, wherein the variances are fixed, the approach retains asymptotic type-I error guarantees.