Bayesian updating of reliability by cross entropy-based importance sampling

Bayesian analysis enables a consistent updating of the failure probability of engineering systems when new data is available. To this end, we introduce an adaptive importance sampling (IS) method based on the principle of cross entropy (CE) minimization. The key contribution is a novel IS density associated with the posterior probability density function (PDF) of the uncertain parameters, that facilitates efficient sampling from the important region of the failure domain, especially when the failure event is rare. The IS density is designed via a two-step procedure. The first step involves construction of a sample-based approximation of the posterior, which we build using the CE method. Here the aim is to determine the parameters of a chosen parametric distribution family that minimize its Kullback-Leibler divergence from the posterior PDF. The second step of the proposed method constructs the desired IS density for sampling the failure domain through a second round of CE minimization, starting from the approximate posterior obtained in the first step. An adaptive, multi-level approach is employed to solve the two CE optimization problems. The IS densities deduced in the two steps are then applied to construct an efficient estimator for the posterior probability of failure. Through numerical studies, we investigate and demonstrate the efficacy of the method in accurately estimating the reliability of engineering systems with rare failure events.