Large-scale administrative or observational datasets are increasingly used to
inform decision making. While this effort aims to ground policy in real-world
evidence, challenges have arise as that selection bias and other forms of
distribution shift often plague observational data. Previous attempts to
provide robust inferences have given guarantees depending on a user-specified
amount of possible distribution shift (e.g., the maximum KL divergence between
the observed and target distributions). However, decision makers will often
have additional knowledge about the target distribution which constrains the
kind of shifts which are possible. To leverage such information, we proposed a
framework that enables statistical inference in the presence of distribution
shifts which obey user-specified constraints in the form of functions whose
expectation is known under the target distribution. The output is
high-probability bounds on the value an estimand takes on the target
distribution. Hence, our method leverages domain knowledge in order to
partially identify a wide class of estimands. We analyze the computational and
statistical properties of methods to estimate these bounds, and show that our
method can produce informative bounds on a variety of simulated and
semisynthetic tasks