论文标题
在稀疏正常手段中，后收缩率和一组收缩先验的渐近贝叶斯最优性问题
Posterior Contraction rate and Asymptotic Bayes Optimality for one-group shrinkage priors in sparse normal means problem
论文作者
论文摘要
我们考虑一个高维稀疏的正常平均值模型，其中的目标是估计假设非零均值的比例是未知的。我们通过一个一组全本网本收缩术对平均向量进行建模，该收缩率属于包括马蹄铁先验的一系列此类先验。我们解决了一些与上述阶级先验的平均向量后载体后验分布的渐近特性有关的问题。我们考虑了两种模拟本文全局参数的方法。首先，将其视为未知的固定参数，然后将其视为经验贝叶斯的估计。在第二种方法中，我们通过为其分配合适的非脱位分布来进行分层贝叶斯处理。我们首先表明，对于正在研究的先验类别，使用经验贝叶斯方法时，平均向量合同的后验分布围绕真实参数的近极速率。接下来，我们证明在分层贝叶斯方法中，相应的贝叶斯估计在平方误差损失函数下渐近地达到了最小风险。我们还表明，围绕真实参数的后验合同以接近最小值的速率。这些结果概括了van der Pas等人的结果。 （2014）\ cite {van2014horseshoe}，（2017）\ cite {van2017appive}，被证明是在马蹄上证明的。我们还研究了这项工作，渐近贝叶斯的最佳性是全球收缩先验的最佳性，在这些研究中，非无效假设的数量尚不清楚。在这里，我们的目标是提出有关全局参数的先前密度的某些条件，以使决策规则引起的贝叶斯风险达到最佳贝叶斯风险，直至某种乘法常数。在Bogdan等人的渐近框架下，使用我们提出的条件。 （2011）\ cite {bogdan2011asmptotic}，我们能够提供一个肯定的答案以满足我们的直觉。
We consider a high-dimensional sparse normal means model where the goal is to estimate the mean vector assuming the proportion of non-zero means is unknown. We model the mean vector by a one-group global-local shrinkage prior belonging to a broad class of such priors that includes the horseshoe prior. We address some questions related to asymptotic properties of the resulting posterior distribution of the mean vector for the said class priors. We consider two ways to model the global parameter in this paper. Firstly by considering this as an unknown fixed parameter and then by an empirical Bayes estimate of it. In the second approach, we do a hierarchical Bayes treatment by assigning a suitable non-degenerate prior distribution to it. We first show that for the class of priors under study, the posterior distribution of the mean vector contracts around the true parameter at a near minimax rate when the empirical Bayes approach is used. Next, we prove that in the hierarchical Bayes approach, the corresponding Bayes estimate attains the minimax risk asymptotically under the squared error loss function. We also show that the posterior contracts around the true parameter at a near minimax rate. These results generalize those of van der Pas et al. (2014) \cite{van2014horseshoe}, (2017) \cite{van2017adaptive}, proved for the horseshoe prior. We have also studied in this work the asymptotic Bayes optimality of global-local shrinkage priors where the number of non-null hypotheses is unknown. Here our target is to propose some conditions on the prior density of the global parameter such that the Bayes risk induced by the decision rule attains Optimal Bayes risk, up to some multiplicative constant. Using our proposed condition, under the asymptotic framework of Bogdan et al. (2011) \cite{bogdan2011asymptotic}, we are able to provide an affirmative answer to satisfy our hunch.
