One of the central quantities of probabilistic seismic risk assessment studies is the fragility curve, which represents the probability of failure of a mechanical structure conditional on a scalar measure derived from the seismic ground motion. Estimating such curves is a difficult task because, for many structures of interest, few data are available and the data are only binary; i.e., they indicate the state of the structure, failure or non-failure. This framework concerns complex equipments such as electrical devices encountered in industrial installations. In order to address this challenging framework a wide range of the methods in the literature rely on a parametric log-normal model. Bayesian approaches allow for efficient learning of the model parameters. However, the choice of the prior distribution has a non-negligible influence on the posterior distribution and, therefore, on any resulting estimate. We propose a thorough study of this parametric Bayesian estimation problem when the data are limited and binary. Using the reference prior theory as a support, we suggest an objective approach for the prior choice. This approach leads to the Jeffreys prior which is explicitly derived for this problem for the first time. The posterior distribution is proven to be proper (i.e., it integrates to unity) with the Jeffreys prior and improper with some classical priors from the literature. The posterior distribution with the Jeffreys prior is also shown to vanish at the boundaries of the parameters domain, so sampling the posterior distribution of the parameters does not produce anomalously small or large values. Therefore, this does not produce degenerate fragility curves such as unit-step functions and the Jeffreys prior leads to robust credibility intervals. The numerical results obtained on two different case studies, including an industrial case, illustrate the theoretical predictions.

翻译：概率地震风险评估研究的核心量之一为易损性曲线，它表示机械结构在地震动标量测度条件下发生失效的概率。在众多结构类型中，由于可用数据稀少且仅包含二元信息（即结构状态仅分为失效与非失效），此类曲线的估计极具挑战。该框架涉及工业设施中的电气设备等复杂装备。为应对这一难题，现有文献中多种方法均采用参数化对数正态模型。贝叶斯方法能够高效学习模型参数，然而先验分布的选择对后验分布及其估计结果具有不可忽视的影响。针对数据有限且为二元的情况，本文对此类参数化贝叶斯估计问题展开深入研究。我们以参考先验理论为基础，提出一种客观的先验选择方法，首次针对该问题明确推导出杰弗里斯先验。研究表明，采用杰弗里斯先验时后验分布是正常的（即归一化积分为1），而采用文献中部分经典先验时后验分布则存在非正常性。此外，采用杰弗里斯先验的后验分布在参数定义域边界处趋近于零，因此参数后验分布的采样不会产生异常小或大的数值，从而避免生成单位阶跃函数等退化易损性曲线，且杰弗里斯先验能提供稳健的置信区间。基于两个不同案例（含工业案例）的数值结果验证了上述理论预测。