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Dirichlet-Laplace priors for optimal shrinkage

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.