  While the traditional viewpoint in machine learning and statistics assumes
training and testing samples come from the same population, practice belies
this fiction. One strategy -- coming from robust statistics and optimization --
is thus to build a model robust to distributional perturbations. In this paper,
we take a different approach to describe procedures for robust predictive
inference, where a model provides uncertainty estimates on its predictions
rather than point predictions. We present a method that produces prediction
sets (almost exactly) giving the right coverage level for any test distribution
in an $f$-divergence ball around the training population. The method, based on
conformal inference, achieves (nearly) valid coverage in finite samples, under
only the condition that the training data be exchangeable. An essential
component of our methodology is to estimate the amount of expected future data
shift and build robustness to it; we develop estimators and prove their
consistency for protection and validity of uncertainty estimates under shifts.
By experimenting on several large-scale benchmark datasets, including Recht et
al.'s CIFAR-v4 and ImageNet-V2 datasets, we provide complementary empirical
results that highlight the importance of robust predictive validity.
