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Suppose we have data generated according to a multivariate normaldistribution with a fixed unknown mean vector that is sparse in the sense ofbeing nearly black. Optimality of Bayes estimates and posterior concentrationproperties in terms of the minimax risk in the $l_2$ norm corresponding to avery general class of continuous shrinkage priors are studied in this work. Theclass of priors considered is rich enough to include a great variety of heavytailed prior distributions, such as, the three parameter beta normal mixtures(including the horseshoe), the generalized double Pareto, the inverse gamma andthe normal-exponential-gamma priors. Assuming that the number of non-zerocomponents of the mean vector is known, we show that the Bayes estimatorscorresponding to this general class of priors attain the minimax risk in the$l_2$ norm (possibly up to a multiplicative constant) and the correspondingposterior distributions contract around the true mean vector at the minimaxoptimal rate for appropriate choice of the global shrinkage parameter.Moreover, we provide conditions for which these posterior distributionscontract around the corresponding Bayes estimates at least as fast as theminimax risk in the $l_2$ norm. We also provide a lower bound to the totalposterior variance for an important subclass of this general class of shrinkagepriors that includes the generalized double Pareto priors with shape parameter$\alpha=0.5$ and the three parameter beta normal mixtures with parameters$a=0.5$ and $b>0$ (including the horseshoe) in particular. The present work isinspired by the recent work of van der Pas et al. (2014) on the posteriorcontraction properties of the horseshoe prior under the present set-up. Weextend their results for this general class of priors and come up with novelunifying proofs which work for a very broad class of one-group continuousshrinkage priors.

Posterior Concentration Properties of a General Class of Shrinkage Priors around Nearly Black Vectors