The Reciprocal Bayesian LASSO

Throughout the course of the paper, we assume that y and X have been centered at 0 so there is no intercept in the model, where y is the n 1 vector of centered responses, X is the n p matrix of standardized regressors, β is the p 1 vector of coefficients to be estimated, and null is the n 1 vector of independent and identically distributed normal errors with mean 0 and variance σ 2 . Compared to traditional penalization functions that are usually symmetric about 0, continuous and nondecreasing in (0,), the rLASSO penalty functions are decreasing in (0,), discontinuous at 0, and converge to infinity when the coefficients approach zero. From a theoretical standpoint, rLASSO shares the same oracle property and same rate of estimation error with other LASSOtype penalty functions. An early reference to this class of models can be found in Song and Liang (2015), with more recent papers focusing on large sample asymptotics, along with computational strategies for frequentist estimation (Shin et al., 2018; Song, 2018). Our approach differs from this line of work in adopting a Bayesian perspective on rLASSO estimation. Ideally, a Bayesian solution can be obtained by placing appropriate priors on the regression coefficients that will mimic the effects of the rLASSO penalty. As apparent from (1), this arises in assuming a prior for β that decomposes as a product of independent inverse Laplace (double exponential) densities: π (β) p null j 1 λ 2β 2 j exp{ λ β j }I { β j null 0 }.