On Bayesian lasso variable selection and the specification of the shrinkage parameter

Abstract

We propose a Bayesian implementation of the lasso regression that accomplishes both shrinkage and variable selection. We focus on the appropriate specification for the shrinkage parameter λ through Bayes factors that evaluate the inclusion of each covariate in the model formulation. We associate this parameter with the values of Pearson and partial correlation at the limits between significance and insignificance as defined by Bayes factors. In this way, a meaningful interpretation of λ is achieved that leads to a simple specification of this parameter. Moreover, we use these values to specify the parameters of a gamma hyperprior for λ. The parameters of the hyperprior are elicited such that appropriate levels of practical significance of the Pearson correlation are achieved and, at the same time, the prior support of λ values that activate the Lindley-Bartlett paradox or lead to over-shrinkage of model coefficients is avoided. The proposed method is illustrated using two simulation studies and a real dataset. For the first simulation study, results for different prior values of λ are presented as well as a detailed robustness analysis concerning the parameters of the hyperprior of λ. In all examples, detailed comparisons with a variety of ordinary and Bayesian lasso methods are presented.