Distinguishing and integrating aleatoric and epistemic variation in uncertainty quantification

Abstract

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.

Novel Mathematical and Computational Techniques for Robust Uncertainty Quantification

“…In the paper [1] we develop an approach that (i) logically distinguishes those aspects of uncertainty that are treated as stochastic variability from other forms of uncertainty, (ii) in cases where a stochastic model is theoretically valid but for which determination of the distribution is not practical, gives bounds for performance measures that are valid for explicitly identified families of distributions, and (iii) is computationally feasible if ordinary uncertainty propagation is feasible. The different forms of uncertainty that are covered by the formulation include: (a) aleatoric with known distribution; (b) aleatoric with partly known distribution (mingled aleatoric and epistemic); (c) epistemic for which one is willing to model by a family of aleatoric uncertainties, and (d) epistemic where one is only willing to place bounds on the uncertainties.…”

“…This optimizing c, which exists and is unique, can be computed using the same techniques used to compute the performance measure itself. We show in [1] that this function has only one local minimum over c G (0,00], and thus the global minimum is easy to compute.…”

“…In [1] two hybrid risk-sensitive measures are introduced, as well as variations. To simplify we mention only the theory developed for the (more useful) form Al = I log ( efx^'M^rtdy).…”

“…In particular, we think of 7 as a nominal distribution of Y, which should be distinguished from a possible true distribution. The risk sensitive functional A*, whose numerical evaluation can be carried out by a variety of methods (in [1] we use polynomial chaos), is based on the nominal distribution. Through the relative entropy duality, it yield various bounds (depending on c) on a families of distributions, with the relative entropy distance the key metric.…”

“…In [1] examples of various types and numerical data is presented, as well as the corresponding theory where all that is assumed regarding the epistemic variables is that they are constrained to some given set. The use of polynomial chaos methods for the evaluation of the risk-sensitive functionals is developed, and explicit relative entropy distances for common families of distributions are listed.…”