A Bayesian Approach to Multiple-Output Quantile Regression

This article presents a Bayesian approach to multiple-output quantile regression. The prior can be elicited as ex-ante knowledge of the distance of the tau-Tukey depth contour to the Tukey median, the first prior of its kind. The parametric model is proven to be consistent and a procedure to obtain confidence intervals is proposed. A proposal for nonparametric multiple-output regression is also presented. These results add to the literature of misspecified Bayesian modeling, consistency, and prior elicitation for nonparametric multivariate modeling. The model is applied to the Tennessee Project Steps to Achieving Resilience (STAR) experiment and finds a joint increase in tau-quantile subpopulations for mathematics and reading scores given a decrease in the number of students per teacher. Supplementary materials for this article are available online.