The Bayesian Bridge

Abstract: We propose the Bayesian bridge estimator for regularized regression and classification. Two key mixture representations for the Bayesian bridge model are developed: (1) a scale mixture of normals with respect to an alpha-stable random variable; and (2) a mixture of Bartlett--Fejer kernels (or triangle densities) with respect to a two-component mixture of gamma random variables. Both lead to MCMC methods for posterior simulation, and these methods turn out to have complementary domains of maximum efficiency. The first representation is a well known result due to West (1987), and is the better choice for collinear design matrices. The second representation is new, and is more efficient for orthogonal problems, largely because it avoids the need to deal with exponentially tilted stable random variables. It also provides insight into the multimodality of the joint posterior distribution, a feature of the bridge model that is notably absent under ridge or lasso-type priors. We prove a theorem that extends this representation to a wider class of densities representable as scale mixtures of betas, and provide an explicit inversion formula for the mixing distribution. The connections with slice sampling and scale mixtures of normals are explored. On the practical side, we find that the Bayesian bridge model outperforms its classical cousin in estimation and prediction across a variety of data sets, both simulated and real. We also show that the MCMC for fitting the bridge model exhibits excellent mixing properties, particularly for the global scale parameter. This makes for a favorable contrast with analogous MCMC algorithms for other sparse Bayesian models. All methods described in this paper are implemented in the R package BayesBridge. An extensive set of simulation results are provided in two supplemental files.