Universal properties of kernel functions for probabilistic sensitivity analysis

Development of probabilistic sensitivities is frequently considered an essential component of a probabilistic analysis and often critical towards understanding the physical mechanisms underlying failure and modifying the design to mitigate and manage risk. One useful sensitivity is the partial derivative of the probability-of-failure and/or the system response with respect to the parameters of the independent input random variables. Calculation of these partial derivatives has been established in terms of an expected value operation (sometimes called the score function or likelihood ratio method). The partial derivatives can be computed with typically insignificant additional computational cost given the failure samples and kernel functions — which are the partial derivatives of the log of the probability density function (PDF) with respect to the parameters of the distribution. The formulation is general such that any sampling method can be used for the computation such as Monte Carlo, importance sampling, Latin hypercube, etc. In this paper, useful universal properties of the kernel functions that must be satisfied for all two parameter independent distributions are derived. These properties are then used to develop distribution-free analytical expressions of the partial derivatives of the response moments (mean and standard deviation) with respect to the PDF parameters for linear and quadratic response functions. These universal properties can be used to facilitate development and verification of the required kernel functions and to develop an improved understanding of the model for design considerations.