One of the crucial quantities of probabilistic seismic risk assessment studies is the fragility curve, which represents the probability of failure of a mechanical structure conditional to a scalar measure derived from the seismic ground motion. Estimating such curves is a difficult task because for most structures of interest, few data are available, whether they come from complex numerical simulations or experimental campaigns. For this reason, a wide range of the methods of the literature rely on a parametric log-normal model. Bayesian approaches allow for efficient learning of the model parameters. However, for small data set sizes, 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 binary (i.e. data indicate the state of the structure, failure or non-failure). Using the reference prior theory as a support, we suggest an objective approach for the prior choice to simulate a posteriori fragility curves. This approach leads to the Jeffreys prior and we prove that this prior depends only of the ground motion characteristics, making its calculation suitable for any equipment in an industrial installation subjected to the same seismic hazard. Our proposal is theoretically and numerically compared to those classically proposed in the literature by considering three different case studies. The results show the robustness and advantages of the Jeffreys prior in terms of regularization (no degenerate estimations) and stability (no outliers of the parameters) for fragility curves estimation.

翻译：概率地震风险评估研究中的关键量之一是易损性曲线，它表示机械结构在给定地震动标量测度下的失效概率。由于大多数感兴趣的结构仅有少量数据可用（无论这些数据来自复杂数值模拟还是实验研究），因此估计此类曲线是一项困难任务。基于此，文献中的大量方法依赖于参数化对数正态模型。贝叶斯方法能够实现模型参数的高效学习，然而在小样本量情况下，先验分布的选择对后验分布及其估计结果具有不可忽略的影响。我们针对数据为二元变量（即指示结构失效或非失效状态）的情况，对这一参数化贝叶斯估计问题开展了深入研究。借助参考先验理论，我们提出了一种用于先验选择的客观方法，以模拟后验易损性曲线。该方法导出了杰弗里斯先验，并证明该先验仅取决于地震动特征，使其适用于遭受相同地震风险的工业设施中任何设备的计算。通过考虑三个不同案例研究，我们将所提方法与文献中经典方法进行了理论和数值对比。结果表明，杰弗里斯先验在易损性曲线估计的正则化（无非退化估计）和稳定性（无参数异常值）方面具有鲁棒性和优势。