  During the past decade, shrinkage priors have received much attention in
Bayesian analysis of high-dimensional data. This paper establishes the
posterior consistency for high-dimensional linear regression with a class of
shrinkage priors, which has a heavy and flat tail and allocates a sufficiently
large probability mass in a very small neighborhood of zero. While enjoying its
efficiency in posterior simulations, the shrinkage prior can lead to a nearly
optimal posterior contraction rate and variable selection consistency as the
spike-and-slab prior. Our numerical results show that under the posterior
consistency, Bayesian methods can yield much better results in variable
selection than the regularization methods such as Lasso and SCAD. This paper
also establishes a Bernstein von-Mises type result, which leads to a convenient
way of uncertainty quantification for regression coefficient estimates.
