%0 Journal Article %1 castillo2015bayesian %A Castillo, Ismaël %A Schmidt-Hieber, Johannes %A van der Vaart, Aad %D 2015 %I Institute of Mathematical Statistics %J The Annals of Statistics %K Bayesian lasso methods sparsification statistics %N 5 %P 1986--2018 %R 10.1214/15-aos1334 %T Bayesian linear regression with sparse priors %U https://doi.org/10.1214%2F15-aos1334 %V 43 %X 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{castillo2015bayesian, 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. }, added-at = {2020-09-18T05:37:56.000+0200}, author = {Castillo, Ismaël and Schmidt-Hieber, Johannes and van der Vaart, Aad}, biburl = {https://www.bibsonomy.org/bibtex/29a3b0ad449e82a1b86bde79530a703a8/peter.ralph}, doi = {10.1214/15-aos1334}, interhash = {72ce9de68dfbd5b7c768bf9f5d6d68ab}, intrahash = {9a3b0ad449e82a1b86bde79530a703a8}, journal = {The Annals of Statistics}, keywords = {Bayesian lasso methods sparsification statistics}, month = oct, number = 5, pages = {1986--2018}, publisher = {Institute of Mathematical Statistics}, timestamp = {2020-09-18T05:37:56.000+0200}, title = {Bayesian linear regression with sparse priors}, url = {https://doi.org/10.1214%2F15-aos1334}, volume = 43, year = 2015 }