## The Annals of Statistics

### Bayesian linear regression with sparse priors

#### Abstract

We study full Bayesian procedures for high-dimensional linear regression under sparsity constraints. The prior is a mixture of point masses at zero and continuous distributions. Under compatibility conditions on the design matrix, the posterior distribution is shown to contract at the optimal rate for recovery of the unknown sparse vector, and to give optimal prediction of the response vector. It is also shown to select the correct sparse model, or at least the coefficients that are significantly different from zero. The asymptotic shape of the posterior distribution is characterized and employed to the construction and study of credible sets for uncertainty quantification.

#### Article information

Source
Ann. Statist., Volume 43, Number 5 (2015), 1986-2018.

Dates
Received: March 2014
Revised: March 2015
First available in Project Euclid: 3 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.aos/1438606851

Digital Object Identifier
doi:10.1214/15-AOS1334

Mathematical Reviews number (MathSciNet)
MR3375874

Zentralblatt MATH identifier
06502640

Subjects
Primary: 62G20: Asymptotic properties
Secondary: 62G05: Estimation

Keywords
Bayesian inference sparsity

#### Citation

Castillo, Ismaël; Schmidt-Hieber, Johannes; van der Vaart, Aad. Bayesian linear regression with sparse priors. Ann. Statist. 43 (2015), no. 5, 1986--2018. doi:10.1214/15-AOS1334. https://projecteuclid.org/euclid.aos/1438606851

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