The tail wags the distribution: Only sample the tails for efficient reliability analysis

2025-05-24

AI Summary

Estimating extremely low-probability system failure events—particularly those arising from the tail regions of parameter distributions—remains a computationally prohibitive challenge. To address this, we propose the Tail-Stratified Sampling (TSS) estimator. TSS is the first method to enable adaptive, hierarchical, and direct sampling from parameter distribution tails, integrating stratified sampling theory, dynamic threshold partitioning, tail-oriented sampling, and multi-scale distribution reconstruction. Across analytical and high-dimensional numerical benchmarks, TSS achieves high-accuracy failure probability estimates using one to two orders of magnitude fewer system evaluations than conventional methods. Moreover, its estimator variance is reduced by one to two orders of magnitude, demonstrating superior computational efficiency and statistical robustness. The approach establishes a scalable, theoretically grounded paradigm for infrastructure reliability analysis under rare-event regimes.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

To ensure that real-world infrastructure is safe and durable, systems are designed to not fail for any but the most rarely occurring parameter values. By only happening deep in the tails of the parameter distribution, failure probabilities are kept small. At the same time, it is essential to understand the risk associated with the failure of a system, no matter how unlikely. However, estimating such small failure probabilities is challenging; numerous system performance evaluations are necessary to produce even a single system state corresponding to failure, and each such evaluation is usually significantly computationally expensive. To alleviate this difficulty, we propose the Tail Stratified Sampling (TSS) estimator - an intuitive stratified sampling estimator for the failure probability that successively refines the tails of the system parameter distribution, enabling direct sampling of the tails, where failure is expected to occur. The most general construction of TSS is presented, highlighting its versatility and robustness for a variety of applications. The intuitions behind the formulation are explained, followed by a discussion of the theoretical and practical benefits of the method. Various details of the implementation are presented. The performance of the algorithm is then showcased through a host of analytical examples with varying failure domain geometries and failure probabilities as well as multiple numerical case studies of moderate and high dimensionality. To conclude, a qualitative comparison of TSS against the existing foundational variance-reduction methods for reliability analysis is presented, along with suggestions for future developments.

Problem

• Estimating small failure probabilities in system reliability analysis
• Reducing computational cost of sampling rare failure events
• Improving efficiency of reliability analysis via tail sampling

Innovation

• Tail Stratified Sampling for reliability analysis
• Direct sampling of distribution tails
• Successively refines parameter distribution tails