Enhancing subset simulation through Bayesian inference

Analyzing and reducing uncertainty in estimating failure probability has always been a crucial part of reliability analysis using Monte Carlo simulation. This paper employs Bayesian inference to capture uncertainty within Subset Simulation (SuS), with a specific emphasis on constructing the posterior distribution of failure probability. Two types of Bayesian models are discussed. The first type is based on the distribution form of the failure probability estimator from the SuS, where the Log-normal and Beta distributions are compared and analyzed. The second type is based on the distribution of statistics in Bernoulli trials in SuS, involving models such as the Beta-Binomial and Uniform-Bernoulli trials with Sequential Dependence. After evaluating all models among two types, the Log-normal distribution-based model was selected as the optimal solution due to its consideration of sample correlation and closed form of Maximum A Posteriori (MAP). Further exploration of this model reveals that its MAP estimation can provide less bias and deviation compared to mean estimation when SuS is run multiple times. Additionally, when compared to the point estimation of failure probability in conventional SuS, the posterior distribution of failure probability provides a more comprehensive overview, capturing the effects of sample size and MCMC algorithms.