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.
翻译:在稀有序列模型中,我们考虑一个受欢迎的贝叶斯多重测试程序,并首次从经常点的角度调查其行为。鉴于在高维稀疏未知参数上之前有一个钉和板之前的钉,人们可以很容易地计算出从钉中产生的后数概率,这与众所周知的当地-fdr值相对应,也称为$-美元价值。峰值和悬浮重量参数以实证贝斯方式校准,使用最小的可能性。正在研究的多重测试程序,这里称为累计美元-价值程序,根据他们的实证值$/ell美元-价值和阈值排列坐标,以便累积的定值不超过用户指定水平美元。我们从多重测试角度验证这一方法的使用:对于适当大信号强度的替代物,该程序的假发现率显示与目标值美元一致,而其假负值率(FNR)为美元。我们通过提供该方法的趋同率来补充这一研究,我们通过提供该方法的趋同率,我们证明这一比值的汇率。