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October 2007 On optimality of Bayesian testimation in the normal means problem
Felix Abramovich, Vadim Grinshtein, Marianna Pensky
Ann. Statist. 35(5): 2261-2286 (October 2007). DOI: 10.1214/009053607000000226


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 l0-type penalties. The penalties are associated with various choices of the prior distributions πn(⋅) 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 πn(⋅). Furthermore, they achieve optimality in a rather general setting under very mild conditions on the prior. We also specify the class of priors πn(⋅) 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.


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Felix Abramovich. Vadim Grinshtein. Marianna Pensky. "On optimality of Bayesian testimation in the normal means problem." Ann. Statist. 35 (5) 2261 - 2286, October 2007.


Published: October 2007
First available in Project Euclid: 7 November 2007

MathSciNet: MR2363971
zbMATH: 1126.62003
Digital Object Identifier: 10.1214/009053607000000226

Primary: 62C10
Secondary: 62C20 , 62G05

Keywords: Adaptivity , complexity penalty , maximum a posteriori rule , minimax estimation , sequence estimation , Sparsity , thresholding

Rights: Copyright © 2007 Institute of Mathematical Statistics


Vol.35 • No. 5 • October 2007
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