Bayesian scale mixtures of normals linear regression and Bayesian quantile regression with big data and variable selection

Abstract

Quantile regression, which estimates various conditional quantiles of a response variable, including the median (0.5th quantile), is particularly useful when the conditional distribution is asymmetric or heterogeneous or fat-tailed or truncated. Bayesian methods for the inference of quantile regression have been receiving increasing attention from both theoretical and empirical viewpoints but facing the challenge of scaling up when the data are too large to be processed by a single machine under many big data environments nowadays. In this paper, we develop a structure link between Bayesian scale mixtures of normals linear regression and Bayesian quantile regression (BQR) via normal-inverse-gamma (NIG) distribution type of likelihood function, prior distribution and posterior distribution. We further explore the detailed methods of BQR for big data, variable selection and posterior prediction. The performance of the proposed techniques is evaluated via simulation studies and a real data analysis.