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 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. 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, 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. 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 Jeffreys' prior and improper with some classical priors from the literature. The posterior distribution with Jeffreys' prior is also shown to vanish at the boundaries of the parameter domain, so sampling of the posterior distribution of the parameters does not produce anomalously small or large values, which in turn does not produce degenerate fragility curves such as unit step functions. The numerical results on three different case studies illustrate these theoretical predictions.

翻译：概率地震风险评估的核心量之一是易损性曲线，它表示机械结构在地震动标量测度条件下失效的概率。由于大多数感兴趣的结构可用数据非常有限，估计此类曲线是一项艰巨任务。因此，文献中广泛采用的方法依赖于参数化对数正态模型。贝叶斯方法能够高效学习模型参数，但先验分布的选择对后验分布及其估计结果存在不可忽视的影响。本文针对二元数据（即指示结构失效或非失效状态的数据）场景，对该参数化贝叶斯估计问题进行了深入研究。基于参考先验理论，我们提出了一种客观的先验选择方法，首次在该问题中显式推导出杰弗里斯先验。证明该先验下的后验分布是正常的（即积分为1），而文献中经典先验会导致非正常后验。此外，杰弗里斯先验下的后验分布在参数域边界趋于零，因此参数后验分布的采样不会产生异常微小或巨大的数值，进而避免生成如单位阶跃函数等退化的易损性曲线。三个不同案例的数值结果验证了上述理论预测。