Uncertainty Quantification and Propagation of Imprecise Probabilities with Copula Dependence Modeling
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Imprecise probability allows for partial probability specifications and is applicable when data is so limited that a unique probability distribution cannot be identified. The primary challenges in imprecise probability relate to quantification of epistemic uncertainty and improving the efficiency of uncertainty propagation with imprecise probabilities - especially for complex systems in high dimensions with dependence among random variables. Zhang and Shields (2018) developed a novel UQ methodology for quantifying and efficiently propagating imprecise probabilities with independent uncertainties resulting from small datasets. In this work, we generalize this novel UQ methodology to overcome the limitations of the independence assumption by modeling the dependence structure using copula theory. The approach uses Bayesian multimodel inference to quantify the copula model uncertainty - determining a set of possible candidate families as well as marginal probability model uncertainty. Parameter uncertainty in copula model and marginal distribution is estimated by Bayesian inference. We then employ importance sampling for efficiently propagating of full uncertainties in dependence modeling. The generalized approach achieves particularly precise estimates for imprecise probabilities with copula dependence modeling for the composite material problem.
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