正态-β素先验下的大规模多假设检验
Large-Scale Multiple Hypothesis Testing with the Normal-Beta Prime Prior
摘要：
我们重新审视的问题，同时测试的手段，$n$独立的正常观测稀疏。我们采取贝叶斯方法来解决这个问题，通过引入一个规模混合先验称为正常β素数（NBP）先验。当将贝塔素密度用作贝叶斯分层模型中的尺度参数时，我们首先推导出新的浓度性质。为了检测数据中的信号，我们提出了一种基于NBP先验下的后验收缩权重阈值的假设检验。以损失函数的预期数量的误分类测试，我们表明，我们的测试过程渐近达到最佳贝叶斯风险时，信号的比例$p$是已知的。当$p$是未知的，我们引入了一个经验贝叶斯变量的测试，也渐近达到贝叶斯甲骨文的风险在整个范围内的稀疏参数$p \propto n^{-ε}, ε\in (0, 1)$。最后，我们还考虑了限制性边际极大似然（REML）和层次贝叶斯方法估计的NBP先验的关键超参数，并在这些框架下检查多重检验。
英文摘要：
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 introducing a scale-mixture prior known as the normal-beta prime (NBP) prior. We first derive new concentration properties when the beta prime density is employed for a scale parameter in Bayesian hierarchical models. To detect signals in our data, we then 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 $p \propto n^{-ε}, ε\in (0, 1)$. 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.
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