One of the key elements of probabilistic seismic risk assessment studies is the fragility curve, which represents the conditional probability of failure of a mechanical structure for a given scalar measure derived from seismic ground motion. For many structures of interest, estimating these curves is a daunting task because of the limited amount of data available; data which is only binary in our framework, i.e., only describing the structure as being in a failure or non-failure state. A large number of methods described in the literature tackle this challenging framework through parametric log-normal models. Bayesian approaches, on the other hand, allow model parameters to be learned more efficiently. However, the impact of the choice of the prior distribution on the posterior distribution cannot be readily neglected and, consequently, neither can its impact on any resulting estimation. This paper proposes a comprehensive study of this parametric Bayesian estimation problem for limited and binary data. Using the reference prior theory as a cornerstone, this study develops an objective approach to choosing the prior. This approach leads to the Jeffreys prior, which is derived for this problem for the first time. The posterior distribution is proven to be proper with the Jeffreys prior but improper with some traditional priors found in the literature. With the Jeffreys prior, the posterior distribution is also shown to vanish at the boundaries of the parameters' domain, which means that sampling the posterior distribution of the parameters does not result in anomalously small or large values. Therefore, the use of the Jeffreys prior does not result in degenerate fragility curves such as unit-step functions, and leads to more robust credibility intervals. The numerical results obtained from different case studies-including an industrial example-illustrate the theoretical predictions.

翻译：概率地震风险评估研究的关键要素之一是易损性曲线，它表示机械结构在地震动标量测度下的失效条件概率。对于许多重要结构而言，由于可用数据有限（且仅包含二元数据，即仅描述结构处于失效或非失效状态），估计这些曲线是一项艰巨任务。文献中大量方法通过参数化对数正态模型处理这一挑战性框架。另一方面，贝叶斯方法能够更高效地学习模型参数。然而，先验分布选择对后验分布的影响不可轻易忽略，因此其任何估计结果的影响也同样无法忽视。本文针对有限二元数据的参数化贝叶斯估计问题提出了系统性研究。以参考先验理论为基础，本研究发展了一种客观的先验选择方法，首次推导出适用于该问题的杰弗里斯先验。证明表明，采用杰弗里斯先验时后验分布是恰当的，而文献中某些传统先验会导致非恰当后验分布。此外，杰弗里斯先验下的后验分布在参数域边界处趋于零，这意味着参数的后验分布采样不会产生异常小或大的值。因此，使用杰弗里斯先验不会导致单位阶跃函数等退化易损性曲线，并能获得更稳健的置信区间。包含工业实例在内的多案例数值分析结果验证了理论预测。