ABSTRACT We revisit the problem of simultaneously testing the means of n independent normal observations under sparsity. We take a Bayesian approach to this problem by studying a scale-mixture prior known as the normal-beta prime (NBP) prior. To detect signals, we propose a hypothesis test based on thresholding the posterior shrinkage weight under the NBP prior. Taking the loss function to be the expected number of misclassified tests, we show that our test procedure asymptotically attains the optimal Bayes risk when the signal proportion p is known. When p is unknown, we introduce an empirical Bayes variant of our test which also asymptotically attains the Bayes Oracle risk in the entire range of sparsity parameters . Finally, we also consider restricted marginal maximum likelihood (REML) and hierarchical Bayes approaches for estimating a key hyperparameter in the NBP prior and examine multiple testing under these frameworks.

中文翻译：

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使用正态β素数先验的大规模多重假设检验

摘要 我们重新审视了在稀疏情况下同时测试 n 个独立正态观测的均值的问题。我们通过研究称为正常β素数（NBP）先验的尺度混合先验，对这个问题采取贝叶斯方法。为了检测信号，我们提出了一个假设检验，该假设检验基于对 NBP 先验下的后收缩权重进行阈值化。将损失函数作为错误分类测试的预期数量，我们表明当信号比例 p 已知时，我们的测试程序渐近地达到最佳贝叶斯风险。当 p 未知时，我们引入了我们测试的经验贝叶斯变体，它也在整个稀疏参数范围内渐近地获得了贝叶斯 Oracle 风险。最后，