The Annals of Statistics
- Ann. Statist.
- Volume 43, Number 5 (2015), 1986-2018.
Bayesian linear regression with sparse priors
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.
Ann. Statist., Volume 43, Number 5 (2015), 1986-2018.
Received: March 2014
Revised: March 2015
First available in Project Euclid: 3 August 2015
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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
- Bayesian linear regression with sparse priors. In the supplement we state a Bernstein–von Mises type result for large lambda and give the remaining proofs.