Posterior Contraction rate for one group global-local shrinkage priors in sparse normal means problem

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. Using a Bayesian setting, 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. Since the global shrinkage parameter plays a pivotal role in capturing the sparsity in the model, we consider two ways to model this parameter in this paper. Firstly, we consider this as an unknown fixed parameter and estimate it by an empirical Bayes estimate. 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 optimality of the horseshoe+ prior to this context. For horseshoe+ prior, we prove that using the empirical Bayes estimate of the global parameter, the corresponding Bayes estimate attains the near minimax risk asymptotically under the squared error loss function and also shows that the posterior distribution contracts around the true parameter at a near minimax rate.