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On optimality of Bayesian testimation in the normal means problem. (English) Zbl 1126.62003
Summary: We consider the problem of recovering a high-dimensional vector \(\mu \) observed in white noise, where the unknown vector \(\mu \) 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 \(\pi_n(\cdot )\) on the number of nonzero entries of \(\mu \) 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 \(\pi_n(\cdot )\). Furthermore, they achieve optimality in a rather general setting under very mild conditions on the prior. We also specify the class of priors \(\pi_n(\cdot )\) 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.

MSC:
62C10 Bayesian problems; characterization of Bayes procedures
62F03 Parametric hypothesis testing
62C20 Minimax procedures in statistical decision theory
62G05 Nonparametric estimation
62F10 Point estimation
Software:
EBayesThresh
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