Statistics and Its Interface

Volume 14 (2021)

Number 4

Prior conditioned on scale parameter for Bayesian quantile LASSO and its generalizations

Pages: 459 – 474

DOI: https://dx.doi.org/10.4310/21-SII666

Authors

Zhongheng Cai (School of Statistics, Faculty of Economics and Management, East China Normal University, Shanghai, China)

Dongchu Sun (School of Statistics, Faculty of Economics and Management, East China Normal University, Shanghai, China; and Department of Statistics, University of Nebraska, Lincoln, Ne., U.S.A.)

Abstract

Several undesirable issues exist in the Bayesian quantile LASSO and its two generalizations, quantile group LASSO and bridge quantile regression (Alhamzawi et al. [1]; Alhamzawi and Algamal [2]; Li et al. [21]). In this paper, we numerically show that, the joint posterior may be multimodal using unconditional prior for the regression coefficients and the posterior estimates may be sensitive to the hyperparameters in Gamma prior frequently used for the scale parameter. We also theoretically illustrate that the joint posterior may be improper when an invariant prior is used for the scale parameter, especially when predictors outnumber observations. To resolve the issues in a unified framework, we propose applying the priors conditioned on the scale parameter for the coefficients along with invariant prior to the scale parameter. We justify the prior choice under one general likelihood including asymmetric Laplace density and the common class of conditioned priors by establishing the corresponding sufficient and necessary condition of the posterior propriety. In addition, we develop ready-to-use partially collapsed Gibbs sampling algorithms for all methods to aid computations. Simulation studies and a real data example demonstrate that our methods usually outperform the original Bayesian approaches.

Keywords

asymmetric Laplace density, Bayesian quantile LASSO, bridge regression, group LASSO, posterior propriety

Received 1 November 2020

Accepted 4 March 2021

Published 8 July 2021