Subset Simulation
Subset simulation estimates rare failure probabilities as a product of larger conditional probabilities reached through intermediate thresholds.
Breaking a rare event into steps
A very small failure probability is hard to sample directly. Subset simulation, introduced by Au and Beck, expresses it as a product: P_f = P(F_1) * P(F_2|F_1) * ... * P(F_m|F_{m-1}), where the intermediate failure regions F_1 superset F_2 superset ... superset F are nested. Each conditional probability is moderate (often around 0.1) and easy to estimate.
Markov chain sampling
Samples in each intermediate region are generated by Monte Carlo at the first level and by Markov chain Monte Carlo (modified Metropolis) at subsequent levels, seeded from the surviving samples of the previous level. The intermediate thresholds are set adaptively so that a fixed fraction of samples pass, keeping each conditional probability well estimated.
Why it scales
- Total cost grows logarithmically in the target probability, not inversely
- Handles high-dimensional input spaces better than importance sampling
- Requires no gradient or design-point search, unlike FORM and SORM
Multiple failure modes
Because it explores the failure region by sampling rather than locating a single point, subset simulation naturally captures multiple disjoint failure modes that trip up FORM. This robustness is a major reason it is a standard reference method for structural reliability.
Cautions
The Markov chains introduce correlation, so the effective sample size is smaller than the nominal count and the estimator variance is larger than independent sampling would suggest. Chain tuning affects accuracy, and poor mixing in high dimension degrades results. Report the coefficient of variation of the estimate, and combine with a surrogate when the underlying model is expensive.