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Summary: The Annals of Statistics
2007, Vol. 35, No. 5, 22612286
DOI: 10.1214/009053607000000226
© Institute of Mathematical Statistics, 2007
ON OPTIMALITY OF BAYESIAN TESTIMATION IN THE
NORMAL MEANS PROBLEM
BY FELIX ABRAMOVICH, VADIM GRINSHTEIN AND MARIANNA PENSKY1
Tel Aviv University, The Open University of Israel and
University of Central Florida
We consider a problem of recovering a high-dimensional vector µ ob-
served 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 n(·) 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 thresh-
olding and model selection procedures as particular cases corresponding to
specific choices of n(·). Furthermore, they achieve optimality in a rather
general setting under very mild conditions on the prior. We also specify the
class of priors n(·) for which the resulting estimator is adaptively optimal
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