On optimality of Bayesian testimation in the normal means problem

Abramovich, Felix; Grinshtein, Vadim; Pensky, Marianna

On optimality of Bayesian testimation in the normal means problem

Authors

Abramovich, Felix

Grinshtein, Vadim

Pensky, Marianna

Description

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.
Published in at http://dx.doi.org/10.1214/009053607000000226 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)