  This paper presents a Bayesian approach to multiple-output quantile
regression. The unconditional model is proven to be consistent and
asymptotically correct frequentist confidence intervals can be obtained. The
prior for the unconditional model can be elicited as the ex-ante knowledge of
the distance of the tau-Tukey depth contour to the Tukey median, the first
prior of its kind. A proposal for conditional regression is also presented. The
model is applied to the Tennessee Project Steps to Achieving Resilience (STAR)
experiment and it finds a joint increase in tau-quantile subpopulations for
mathematics and reading scores given a decrease in the number of students per
teacher. This result is consistent with, and much stronger than, the result one
would find with multiple-output linear regression. Multiple-output linear
regression finds the average mathematics and reading scores increase given a
decrease in the number of students per teacher. However, there could still be
subpopulations where the score declines. The multiple-output quantile
regression approach confirms there are no quantile subpopulations (of the
inspected subpopulations) where the score declines. This is truly a statement
of `no child left behind' opposed to `no average child left behind.'
