Improved variance estimation for subset simulation by accounting for the correlation between Markov chains

Subset simulation (SS) is a popular structural reliability analysis method, especially for problems characterized by low failure probabilities and high-dimensional complexities. Unlike most variance reduction methods, SS obviates the need for prior domain information, making it versatile across diverse applications. Markov chain Monte Carlo (MCMC) algorithms are required for sampling from an unknown conditional distribution, resulting in correlated samples. There is plenty of literature on SS in several aspects, such as the improvement of MCMC algorithms, and combining SS with other techniques. However, one aspect that appears to be neglected concerns the variance estimation crucial for assessing the accuracy of the probability estimate. To date, most studies on SS still rely on the conventional variance estimation method, which only considers the correlation within a Markov chain (intrachain) but neglects the correlation across separate chains (interchain) and different subset levels (interlevel). This study aims to improve understanding of this topic and develop a more accurate variance estimation method for SS. An investigation based on multiple independent SS runs reveal that the intrachain, interchain and interlevel correlations are all important. Subsequently, a new variance estimation method is proposed to account for the intrachain and interchain correlations. The proposed method is easy to apply, has small sampling uncertainty and only utilizes samples from a single SS run. Results indicate a notable improvement in accuracy compared to the conventional method.