## Abstract

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.

Original language | English |
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Pages (from-to) | 2261-2286 |

Number of pages | 26 |

Journal | Annals of Statistics |

Volume | 35 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2007 |

## Keywords

- Adaptivity
- Complexity penalty
- Maximum a posteriori rule
- Minimax estimation
- Sequence estimation
- Sparsity
- Thresholding