  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.
