  Quantile regression is a fundamental problem in statistical learning
motivated by a need to quantify uncertainty in predictions, or to model a
diverse population without being overly reductive. For instance,
epidemiological forecasts, cost estimates, and revenue predictions all benefit
from being able to quantify the range of possible values accurately. As such,
many models have been developed for this problem over many years of research in
statistics, machine learning, and related fields. Rather than proposing yet
another (new) algorithm for quantile regression we adopt a meta viewpoint: we
investigate methods for aggregating any number of conditional quantile models,
in order to improve accuracy and robustness. We consider weighted ensembles
where weights may vary over not only individual models, but also over quantile
levels, and feature values. All of the models we consider in this paper can be
fit using modern deep learning toolkits, and hence are widely accessible (from
an implementation point of view) and scalable. To improve the accuracy of the
predicted quantiles (or equivalently, prediction intervals), we develop tools
for ensuring that quantiles remain monotonically ordered, and apply conformal
calibration methods. These can be used without any modification of the original
library of base models. We also review some basic theory surrounding quantile
aggregation and related scoring rules, and contribute a few new results to this
literature (for example, the fact that post sorting or post isotonic regression
can only improve the weighted interval score). Finally, we provide an extensive
suite of empirical comparisons across 34 data sets from two different benchmark
repositories.
