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.
Publié le : 2007-10-14
Classification:
Adaptivity,
complexity penalty,
maximum a posteriori rule,
minimax estimation,
sequence estimation,
sparsity,
thresholding,
62C10,
62C20,
62G05
@article{1194461730,
author = {Abramovich, Felix and Grinshtein, Vadim and Pensky, Marianna},
title = {On optimality of Bayesian testimation in the normal means problem},
journal = {Ann. Statist.},
volume = {35},
number = {1},
year = {2007},
pages = { 2261-2286},
language = {en},
url = {http://dml.mathdoc.fr/item/1194461730}
}
Abramovich, Felix; Grinshtein, Vadim; Pensky, Marianna. On optimality of Bayesian testimation in the normal means problem. Ann. Statist., Tome 35 (2007) no. 1, pp. 2261-2286. http://gdmltest.u-ga.fr/item/1194461730/