Robust bounds on risk-sensitive functionals via rényi divergence

We extend the duality between exponential integrals and relative entropy to a variational formula for exponential integrals involving the Rényi divergence. This formula characterizes the dependence of risk-sensitive functionals to perturbations in the underlying distribution. It also shows that perturbations of related quantities determined by tail behavior, such as probabilities of rare events, can be bounded in terms of the Rényi divergence. The characterization gives rise to tight upper and lower bounds that are meaningful for all values of a large deviation scaling parameter, allowing one to quantify in explicit terms the robustness of risk-sensitive costs. As applications we consider problems of uncertainty quantification when aspects of the model are not fully known, as well their use in bounding tail properties of an intractable model in terms of a tractable one.