On optimality of Bayesian testimation in the normal means problem

Abstract

We consider a problem of recovering a high-dimensional vector $μ$ observed in white noise, where the unknown vector $μ$ is assumed to be sparse. The objective of the paper is to develop a Bayesian formalism which gives rise to a family of $l_0$-type penalties. The penalties are associated with various choices of the prior distributions $π_n(\cdot)$ on the number of nonzero entries of $μ$ and, hence, are easy to interpret. The resulting Bayesian estimators lead to a general thresholding rule which accommodates many of the known thresholding and model selection procedures as particular cases corresponding to specific choices of $π_n(\cdot)$. Furthermore, they achieve optimality in a rather general setting under very mild conditions on the prior. We also specify the class of priors $π_n(\cdot)$ for which the resulting estimator is adaptively optimal (in the minimax sense) for a wide range of sparse sequences and consider several examples of such priors.