Reference prior for Bayesian estimation of seismic fragility curves

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.