  Conformal prediction is a popular tool for providing valid prediction sets
for classification and regression problems, without relying on any
distributional assumptions on the data. While the traditional description of
conformal prediction starts with a nonconformity score, we provide an alternate
(but equivalent) view that starts with a sequence of nested sets and calibrates
them to find a valid prediction set. The nested framework subsumes all
nonconformity scores, including recent proposals based on quantile regression
and density estimation. While these ideas were originally derived based on
sample splitting, our framework seamlessly extends them to other aggregation
schemes like cross-conformal, jackknife+ and out-of-bag methods. We use the
framework to derive a new algorithm (QOOB, pronounced cube) that combines four
ideas: quantile regression, cross-conformalization, ensemble methods and
out-of-bag predictions. We develop a computationally efficient implementation
of cross-conformal, that is also used by QOOB. In a detailed numerical
investigation, QOOB performs either the best or close to the best on all
simulated and real datasets. Code for QOOB is available at
https://github.com/aigen/QOOB.
