  Penalized regression methods, such as $L_1$ regularization, are routinely
used in high-dimensional applications, and there is a rich literature on
optimality properties under sparsity assumptions. In the Bayesian paradigm,
sparsity is routinely induced through two-component mixture priors having a
probability mass at zero, but such priors encounter daunting computational
problems in high dimensions. This has motivated an amazing variety of
continuous shrinkage priors, which can be expressed as global-local scale
mixtures of Gaussians, facilitating computation. In sharp contrast to the
frequentist literature, little is known about the properties of such priors and
the convergence and concentration of the corresponding posterior distribution.
In this article, we propose a new class of Dirichlet--Laplace (DL) priors,
which possess optimal posterior concentration and lead to efficient posterior
computation exploiting results from normalized random measure theory. Finite
sample performance of Dirichlet--Laplace priors relative to alternatives is
assessed in simulated and real data examples.
