Much of uncertainty quantification to date has focused on determining the effect of variables modeled probabilistically, and with a known distribution, on some physical or engineering system. We develop methods to obtain information on the system when the distributions of some variables are known exactly, others are known only approximately, and perhaps others are not modeled as random variables at all. The main tool used is the duality between risk-sensitive integrals and relative entropy, and we obtain explicit bounds on standard performance measures (variances, exceedance probabilities) over families of distributions whose distance from a nominal distribution is measured by relative entropy. The evaluation of the risk-sensitive expectations is based on polynomial chaos expansions, which help keep the computational aspects tractable.

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.

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