Bayesian Quantile Regression

Abstract

This chapter provides a review of Bayesian quantile regression methods based on different types of working likelihoods. It discusses some developments in Bayesian quantile regression approaches based on various working likelihoods, including parametric likelihood based on the asymmetric Laplace distribution, empirical likelihood, and some semiparametric and nonparametric likelihoods. Bayesian approaches provide convenient alternative inference tools for quantile regression. Bayesian quantile regression has several advantages compared to frequentist approaches. Bayesian quantile regression is an important topic that requires better understanding from both the computational and theoretical perspectives. The main challenge for Bayesian quantile regression is that the likelihood has no parametric forms, so a working likelihood is needed for the Bayesian approaches to work. The Bayesian empirical likelihood is a promising approach especially for simultaneous analysis of multiple quantiles, but additional work is needed to adapt the idea to more complex situations such as for censored and longitudinal data.