稀疏序列的经验贝叶斯累积$\ell$值多重检验方法
Empirical Bayes cumulative $\ell$-value multiple testing procedure for sparse sequences
摘要：
在稀疏序列模型中，我们考虑一个流行的贝叶斯多重检验程序，并首次从频率论的角度研究其行为。给定在高维稀疏未知参数上的尖峰和厚片先验，可以容易地计算来自尖峰的后验概率，其对应于众所周知的局部fdr值，也称为$\ell$值。钉和板的重量参数校准的经验贝叶斯的方式，使用边际最大似然。所研究的多重检验程序（此处称为累积$\ell$值程序）根据经验$\ell$值和阈值对坐标进行排序，以使累积排序和不超过用户指定的水平$t$。我们验证了使用这种方法从多个测试的角度来看：适当的大信号强度的替代品，假发现率（FDR）的程序收敛到目标水平$t$，而其假阴性率（FNR）去$0$。我们补充这项研究，提供收敛速度的方法。此外，我们证明了$q$值多重检验程序在该模型中具有相似的收敛速度。
英文摘要：
In the sparse sequence model, we consider a popular Bayesian multiple testing procedure and investigate for the first time its behaviour from the frequentist point of view. Given a spike-and-slab prior on the high-dimensional sparse unknown parameter, one can easily compute posterior probabilities of coming from the spike, which correspond to the well known local-fdr values, also called $\ell$-values. The spike-and-slab weight parameter is calibrated in an empirical Bayes fashion, using marginal maximum likelihood. The multiple testing procedure under study, called here the cumulative $\ell$-value procedure, ranks coordinates according to their empirical $\ell$-values and thresholds so that the cumulative ranked sum does not exceed a user-specified level $t$. We validate the use of this method from the multiple testing perspective: for alternatives of appropriately large signal strength, the false discovery rate (FDR) of the procedure is shown to converge to the target level $t$, while its false negative rate (FNR) goes to $0$. We complement this study by providing convergence rates for the method. Additionally, we prove that the $q$-value multiple testing procedure shares similar convergence rates in this model.
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