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：我们考虑一个高维稀疏正态均值模型，其目标是在非零均值比例未知的情况下估计均值向量。我们通过一个属于包含马蹄形先验在内的广泛先验类别的单组全局-局部收缩先验对均值向量进行建模。针对所述先验类别下均值向量后验分布的渐近性质，我们探讨了若干问题。本文通过两种方式对全局参数进行建模：首先将其视为未知固定参数，随后采用经验贝叶斯方法进行估计；其次通过分配适当的非退化先验分布进行层次贝叶斯处理。我们首先证明，在采用经验贝叶斯方法时，所研究的先验类别下均值向量的后验分布以接近极小极大最优速率向真实参数收缩。随后证明，在层次贝叶斯方法下，对应的贝叶斯估计在平方误差损失函数下渐近达到极小极大风险。同时证明后验分布以接近极小极大最优速率向真实参数收缩。这些结果将van der Pas等人（2014）\cite{van2014horseshoe}、（2017）\cite{van2017adaptive}针对马蹄形先验的结论进行了推广。本文还研究了非零假设数量未知时全局-局部收缩先验的渐近贝叶斯最优性。我们的目标是为全局参数的先验密度提出若干条件，使得决策规则诱导的贝叶斯风险在乘法常数范围内逼近最优贝叶斯风险。基于所提条件，在Bogdan等人（2011）\cite{bogdan2011asymptotic}的渐近框架下，我们能够为这一猜想提供肯定性答案。