An investigation into Markov chain Monte Carlo algorithms for Subset simulation: Emphasizing uncertainty analysis

Subset simulation (SuS) has emerged as a crucial method for estimating failure probability, striking an efficient balance between accuracy and computational cost. This paper introduces an uncertainty analysis method for the Markov chain Monte Carlo (MCMC) algorithm within the SuS framework by examining three key aspects: Effective Sample Size (ESS) ratio, intra-chain correlation, and inter-chain correlation during SuS iterations. The first two affect the standard deviation and bias of the results when MCMC algorithms are run robustly, respectively, while the latter indicates whether the sample distribution generated from MCMC algorithms is stationary or not. Based on these insights, three MCMC algorithms are discussed: Component-wise adaptive Metropolis-Hastings (CW-aMH), Aadaptive Conditional sampling (aCS), and Elliptical Slice sampling (ES). It is demonstrated that better control of the low decreasing rate of ESS ratio in aCS results in a smaller standard deviation compared to CW-aMH in estimation. On the other hand, ES offers better unbiasedness and robustness owing to its low intra-chain correlation and inter-chain correlation. This method provides new insights into MCMC uncertainty analysis, establishing a potential statistical tool for future research in this area.