  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
corresponding frequentist literature, very little is known about the properties
of such priors. Focusing on a broad class of shrinkage priors, we provide
precise results on prior and posterior concentration. Interestingly, we
demonstrate that most commonly used shrinkage priors, including the Bayesian
Lasso, are suboptimal in high-dimensional settings. A new class of Dirichlet
Laplace (DL) priors are proposed, which are optimal 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 simulations.
