Importance sampling for augmented failure probability updating under output observations

When prior sample data are limited and the distribution parameters (DP) of random inputs exhibit uncertainty, Bayesian theory can be utilized to update the distribution of DPs and the augmented failure probability (AFP) integrating newly collected output observations. A critical challenge in this context is the estimation of the likelihood function, which involves the probabili ty density function (PDF) of model outputs respected to DPs realizations. However, existing methods for efficiently addressing this challenge remain inadequate. To overcome this limitation, an efficient failure probability function (FPF)-based approach is proposed to estimate the output PDF and iteratively update the AFP as new output observations become available. In the proposed method, the output PDF is obtained through the derivative of the FPF with respect to output thresholds and DP realizations. To further enhance computational efficiency, an adaptive layered mixture importance sampling technique is developed to compute the FPF. The specialized strategy is introduced for estimating the FPF across varying output thresholds and DP realizations, thereby reducing estimation variance. Illustrative examples validate that the efficiency of estimating FPF and updating AFP is improved by the proposed method.