Bayesian linear regression with sparse priors

Ismaël Castillo, Johannes Schmidt-Hieber, Aad van der Vaart

The Annals of Statistics · 2015 · Volume 43 · Issue 5

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.

Keywords

62G05, 62G20, Bayesian inference, Sparsity

Enable JavaScript for citation tools and additional paper interactions.