This paper presents an extended polynomial chaos formalism for epistemic uncertainties and a new framework for evaluating sensitivities and variations of output probability density functions (PDF) to uncertainty in probabilistic models of input variables. An ”extended” polynomial chaos expansion (PCE) approach is developed that accounts for both aleatory and epistemic uncertainties, modeled as random variables, thus allowing a unified treatment of both types of uncertainty. We explore in particular epistemic uncertainty associated with the choice of prior probabilistic models for input parameters. A PCE-based Kernel Density (KDE) construction provides a composite map from the PCE coefficients and germ to the PDF of quantities of interest (QoI). The sensitivities of these PDF with respect to the input parameters are then evaluated. Input parameters of the probabilistic models are considered. By sampling over the epistemic random variable, a family of PDFs is generated and the failure probability is itself estimated as a random variable with its own PCE. Integrating epistemic uncertainties within the PCE framework results in a computationally efficient paradigm for propagation and sensitivity evaluation. Two typical illustrative examples are used to demonstrate the proposed approach.

本文提出了一种针对认知不确定性的扩展多项式混沌形式主义，并提出了一种新的框架，用于评估输入变量概率模型中输出概率密度函数（PDF）对不确定性的敏感性和变化。开发了一种“扩展”多项式混沌扩展（PCE）方法，该方法考虑了随机变量和建模变量的不确定性和认知不确定性，因此可以对两种类型的不确定性进行统一处理。我们特别探讨与输入参数的先前概率模型的选择相关的认知不确定性。基于PCE的内核密度（KDE）构造提供了来自PCE系数和对PDF的关注数量（QoI）的细菌。然后评估这些PDF相对于输入参数的敏感性。考虑概率模型的输入参数。通过对认知随机变量进行采样，生成了PDF族，并且通过其自身的PCE将故障概率本身估计为随机变量。在PCE框架中整合认知不确定性会导致传播和敏感性评估的计算有效范例。两个典型的说明性示例用于说明所提出的方法。