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.

翻译：我们考虑了一个高维稀疏正常值模型, 目标是估算平均值矢量, 假设非零值的比例未知。 我们用一个组的全局- 全球- 本地缩水模型来模拟平均值矢量, 之前属于包括马铃薯在内的大类前端。 我们处理一些与该类前端平均矢量分布结果的无症状特性有关的问题。 我们考虑两个方法来模拟本文中的全球参数。 首先, 将此值视为一个未知的固定参数, 然后用一个经验性的 Bayes 估计。 在第二个方法中, 我们用一个合适的全球级的平面度/ 本地降水量来模拟平均矢量。 我们首先显示, 对于正在研究中的前端, 当使用实验性的 Bayes 方法时, 平均矢量分布在真实值上, 相应的Bayes 估计在最低值20 中, 在平面错误损失功能下, 将最小值的基底部- 度- 直流- 直径的亚马路路路面- 之前的测算法, 提供了我们当前最前的常数。