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。我们采用属于广泛单组全局-局部收缩先验类（包含马蹄先验）的模型对均值向量进行建模。针对该先验类，我们探讨了所得均值向量后验分布渐近性质的相关问题。本文考虑了两种全局参数的建模方式：首先将其视为未知固定参数并通过经验贝叶斯方法估计；其次采用分层贝叶斯处理，为其赋予合适的非退化先验分布。我们首先证明，当使用经验贝叶斯方法时，所研究先验类下的均值向量后验分布能以近乎极小极大速率收缩至真实参数。接着证明在分层贝叶斯方法中，相应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，同时后验分布也以近乎极小极大速率收缩。这些结果推广了van der Pas等人(2014)\cite{van2014horseshoe}、(2017)\cite{van2017adaptive}针对马蹄先验证明的结论。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，旨在提出全局参数先验密度需满足的条件，使得决策规则诱导的贝叶斯风险达到最优贝叶斯风险（至多相差常数倍）。基于Bogdan等人(2011)\cite{bogdan2011asymptotic}的渐近框架，我们提出的条件为验证该猜想提供了肯定性答案。