  Heavy tailed distributions present a tough setting for inference. They are
also common in industrial applications, particularly with Internet transaction
datasets, and machine learners often analyze such data without considering the
biases and risks associated with the misuse of standard tools. This paper
outlines a procedure for inference about the mean of a (possibly conditional)
heavy tailed distribution that combines nonparametric analysis for the bulk of
the support with Bayesian parametric modeling -- motivated from extreme value
theory -- for the heavy tail. The procedure is fast and massively scalable. The
resulting point estimators attain lowest-possible error rates and, unique among
alternatives, we are able to provide accurate uncertainty quantification for
these estimators. The work should find application in settings wherever correct
inference is important and reward tails are heavy; we illustrate the framework
in causal inference for A/B experiments involving hundreds of millions of users
of eBay.com.
