Robust a posteriori estimation of probit-lognormal seismic fragility curves via sequential design of experiments and constrained reference prior

Seismic fragility curves express the probability of failure of a mechanical equipment conditional to an intensity measure derived from a seismic signal. Although based on a strong assumption, the probit-lognormal model is very popular among practitioners for estimating such curves, judging by its abundant use in the literature. However, as this model is likely to lead to biased estimates, its use should be limited to cases for which only few data are available. In practice, this involves having to resort to binary data which indicate the state of the structure when it has been subjected to a seismic loading, namely failure or non-failure. The question then arises of the choice of data that must be used to obtain an optimal estimate, that is to say the most precise possible with the minimum of data. To answer this question, we propose a methodology for design of experiments in a Bayesian framework based on the reference prior theory. This theory aims to define a so-called objective prior that favors data learning, which is slighty constrained in this work in order tackle the problems of likelihood degeneracy that are ubiquitous with small data sets. The novelty of our work is then twofold. First, we rigorously present the problem of likelihood degeneracy which hampers frequentist approaches such as the maximum likelihood estimation. Then, we propose our strategy inherited from the reference prior theory to build the data set. This strategy aims to maximize the impact of the data on the posterior distribution of the fragility curve. Our method is applied to a case study of the nuclear industry. The results demonstrate its ability to efficiently and robustly estimate the fragility curve, and to avoid degeneracy even with a limited number of experiments. Additionally, we demonstrate that the estimates quickly reach the model bias induced by the probit-lognormal modeling.