Failure probability estimation with failure samples: An extension of the two-stage Markov chain Monte Carlo simulation

In structural reliability analysis, the so-called optimal sampling density is a useful density for failure probability estimation based on importance sampling. This density is also known as the failure-conditional density, which is useful for reliability sensitivity analysis. The recent two-stage Markov chain Monte Carlo (MCMC) simulation is a highly efficient approach to sample from the failure-conditional density and has been used for reliability sensitivity analysis. In this work, the two-stage MCMC simulation is extended for failure probability estimation only with the obtained failure samples. With this extension, both tasks (reliability sensitivity analysis and failure probability estimation) can be achieved from a single set of samples. Two approaches, i.e., the traditional importance sampling equations and the re-targeted harmonic mean, are adopted for failure probability estimation. In both approaches, a density close to the optimal sampling density is required, and the Gaussian mixture (GM) model with cross-entropy is adopted to obtain this density from the available samples. For high-dimensional problems, a one-dimensional update strategy for estimating the covariance matrix of the GM model is adopted to improve failure probability estimation. Several examples are used to test the two approaches, and it shows that the re-targeted harmonic mean approach can provide good failure probability estimation.