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，其目标是在已知非零均值比例未知的情况下估计均值向量。我们采用属于一大类包含马蹄形先验的单组全局-局部收缩先验来建模均值向量，并针对该类先验下均值向量后验分布的渐近性质展开研究。本文采用两种方式处理全局参数：第一种将其视为未知固定参数并通过经验贝叶斯估计获取；第二种采用分层贝叶斯方法为其赋予合适的非退化先验分布。我们首先证明，对于所研究的一类先验，当采用经验贝叶斯方法时，均值向量的后验分布以接近极小化极大速率收缩至真实参数。接着证明在分层贝叶斯方法中，相应的贝叶斯估计在平方误差损失函数下渐近达到极小化极大风险，同时后验分布仍以接近极小化极大速率收缩至真实参数。这些结果推广了van der Pas等人(2014)\cite{van2014horseshoe}、(2017)\cite{van2017adaptive}针对马蹄形先验的结论。本研究还探讨了非零假设数目未知时全局-局部收缩先验的渐近贝叶斯最优性，目标是在全局参数的先验密度上提出若干条件，使决策规则诱导的贝叶斯风险在乘法常数意义下达到最优贝叶斯风险。基于该条件，我们在Bogdan等人(2011)\cite{bogdan2011asymptotic}的渐近框架下给出了支持我们猜想的肯定性答案。