  Many existing approaches for estimating parameters in settings with
distributional shifts operate under an invariance assumption. For example,
under covariate shift, it is assumed that p(y|x) remains invariant. We refer to
such distribution shifts as sparse, since they may be substantial but affect
only a part of the data generating system. In contrast, in various real-world
settings, shifts might be dense. More specifically, these dense distributional
shifts may arise through numerous small and random changes in the population
and environment. First, we will discuss empirical evidence for such random
dense distributional shifts and explain why commonly used models for
distribution shifts-including adversarial approaches-may not be appropriate
under these conditions. Then, we will develop tools to infer parameters and
make predictions for partially observed, shifted distributions. Finally, we
will apply the framework to several real-world data sets and discuss
diagnostics to evaluate the fit of the distributional uncertainty model.
