Elastic Net Regression Modeling With the Orthant Normal Prior

The elastic net procedure is a form of regularized optimization for linear regression that provides a bridge between ridge regression and the lasso. The estimate that it produces can be viewed as a Bayesian posterior mode under a prior distribution implied by the form of the elastic net penalty. This article broadens the scope of the Bayesian connection by providing a complete characterization of a class of prior distributions that generate the elastic net estimate as the posterior mode. The resulting model-based framework allows for methodology that moves beyond exclusive use of the posterior mode by considering inference based on the full posterior distribution. Two characterizations of the class of prior distributions are introduced: a properly normalized, direct characterization, which is shown to be conjugate for linear regression models, and an alternate representation as a scale mixture of normal distributions. Prior distributions are proposed for the regularization parameters, resulting in an infinite mixture of elastic net regression models that allows for adaptive, data-based shrinkage of the regression coefficients. Posterior inference is easily achieved using Markov chain Monte Carlo (MCMC) methods. Uncertainty about model specification is addressed from a Bayesian perspective by assigning prior probabilities to all possible models. Corresponding computational approaches are described. Software for implementing the MCMC methods described in this article, written in C++ with an R package interface, is available at http://www.stat.osu.edu/~hans/software/.