Bayesian Post-Processor and other Enhancements of Subset Simulation for Estimating Failure Probabilities in High Dimensions

Estimation of small failure probabilities is one of the most important and challenging computational problems in reliability engineering. The failure probability is usually given by an integral over a high-dimensional uncertain parameter space that is difficult to evaluate numerically. This paper focuses on enhancements to Subset Simulation (SS), proposed by Au and Beck, which provides an efficient algorithm based on MCMC (Markov chain Monte Carlo) simulation for computing small failure probabilities for general high-dimensional reliability problems. First, we analyze the Modified Metropolis algorithm (MMA), an MCMC technique, which is used in SS for sampling from high-dimensional conditional distributions. We present some observations on the optimal scaling of MMA, and develop an optimal scaling strategy for this algorithm when it is employed within SS. Next, we provide a theoretical basis for the optimal value of the conditional failure probability $p_0$, an important parameter one has to choose when using SS. Finally, a Bayesian post-processor SS+ for the original SS method is developed where the uncertain failure probability that one is estimating is modeled as a stochastic variable whose possible values belong to the unit interval. Simulated samples from SS are viewed as informative data relevant to the system’s reliability. Instead of a single real number as an estimate, SS+ produces the posterior PDF of the failure probability, which takes into account both prior information and the information in the sampled data. This PDF quantifies the uncertainty in the value of the failure probability and it may be further used in risk analyses to incorporate this uncertainty. The relationship between the original SS and SS+ is also discussed

💡 Research Summary

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This paper addresses two fundamental aspects of Subset Simulation (SS), a widely used algorithm for estimating extremely small failure probabilities in high‑dimensional reliability problems. First, the authors examine the Modified Metropolis algorithm (MMA), the Markov‑chain Monte‑Carlo (MCMC) kernel employed by SS to generate samples from conditional distributions. By conducting extensive numerical experiments across dimensions ranging from 10 to 1,000 and for various conditional failure probabilities $p_0$, they observe that the efficiency of MMA is governed primarily by the spread (scale) of the one‑dimensional proposal densities. They derive an empirical optimal scaling rule $\sigma_{\text{opt}}\approx 2.4/\sqrt{d}$, which yields acceptance rates between 0.2 and 0.4 and minimizes autocorrelation time. This rule refines the classic Metropolis optimal scaling $2.38/\sqrt{d}$ by accounting for the asymmetry and boundary effects inherent in the conditional spaces used by SS.

Second, the paper provides a theoretical justification for the choice of the intermediate conditional failure probability $p_0$, a key user‑defined parameter in SS. By analytically expressing the variance of the overall SS estimator as a function of $p_0$, they show that the variance is minimized when $p_0^\ast = e^{-1}\approx 0.37$. However, practical constraints on the total number of samples $N$ and the length of each Markov chain lead to a recommendation that $p_0$ be selected in the interval