On optimality of Bayesian testimation in the normal means problem

By Felix Abramovich, V. Grinshtein, Marianna Pensky. 2007

57 citations recorded by the source corpus.

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 (.) 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 (.). 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 (.) 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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2bde47cee0a40a3601a509500751982c4310e652

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