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

A seismic fragility curve expresses the probability of failure of a structure conditional to an intensity measure (IM) derived from seismic signals. When only limited data is available, the practitioner often refers to the probit-lognormal model coupled with maximum likelihood estimation (MLE) to obtain estimates of these curves. This means that only a binary indicator of the state (BIS) of the structure is known, namely a failure or non-failure state indicator, when it is subjected to a seismic signal with an intensity measure IM. In this context, the objective of this work is to propose a method for optimally estimating such curves by obtaining the most precise estimate possible with the minimum of data. The novelty of our work is twofold. First, we present and show how to mitigate the likelihood degeneracy problem which is ubiquitous with small data sets and hampers frequentist approaches such as MLE. Second, we propose a novel strategy for sequential design of experiments (DoE) that selects seismic signals from a large database of synthetic or real signals via their IM values, to be applied to structures to evaluate the corresponding BISs. This strategy relies on a criterion based on information theory in a Bayesian framework. It therefore aims to sequentially designate the IM value such that the pair (IM, BIS) has on average, with respect to the BIS of the structure, the greatest impact on the posterior distribution of the fragility curve. The methodology is applied to a case study from the nuclear industry. The results demonstrate its ability to efficiently and robustly estimate the fragility curve, and to avoid degeneracy even with a limited amount of data, i.e., less than 100. Furthermore, we demonstrate that the estimates quickly reach the model bias induced by the probit-lognormal modeling. Eventually, two criteria are suggested to help the user stop the DoE algorithm.

💡 Research Summary

This paper addresses the challenge of estimating seismic fragility curves when only a small number of binary observations (failure or non‑failure) are available for a structure subjected to seismic excitations. The authors focus on the widely used probit‑lognormal model, which characterizes a fragility curve by two parameters: the median intensity measure (α) and the log‑standard deviation (β). Traditional maximum‑likelihood estimation (MLE) often fails with limited data because the likelihood can become degenerate, leading to unrealistic step‑function curves or even an improper posterior that cannot be normalized for Bayesian inference.

To overcome these issues, the authors adopt a Bayesian framework based on the theory of reference priors. Specifically, they employ Jeffreys’ prior, derived by maximizing the expected mutual information between data and parameters, which depends only on the distribution of the intensity measure (IM). Since Jeffreys’ prior can be improper, they introduce a modest constraint that renders the prior proper while preserving its objective, data‑driven nature. This constrained reference prior guarantees a proper posterior even when the data set contains no failures or is otherwise sparse.

The core contribution is a sequential design of experiments (DoE) strategy that selects the most informative IM values from a large repository of synthetic or recorded seismic signals. For each candidate IM, the expected impact on the posterior is quantified by the Kullback‑Leibler (KL) divergence between the current posterior and the posterior that would result after observing a new binary outcome at that IM. The IM that maximizes this expected KL divergence is chosen for the next test, the binary response is recorded, and the posterior is updated via Markov Chain Monte Carlo (MCMC). This information‑theoretic criterion ensures that each additional experiment provides the greatest possible reduction in parameter uncertainty.

To make the algorithm computationally tractable, the authors derive an analytical approximation of Jeffreys’ prior and a closed‑form expression for the expected KL divergence, allowing rapid evaluation of many candidate IMs. They also propose two practical stopping rules: (1) when the posterior variance of β (or α) falls below a user‑defined threshold, indicating sufficient certainty, and (2) when the incremental expected KL gain becomes negligible, suggesting diminishing returns from further testing.

The methodology is validated on a nuclear‑industry piping system. A stochastic signal generator calibrated on 97 real accelerograms produces 10⁵ synthetic ground motions covering magnitudes 5.5–6.5 and distances <20 km. A low‑fidelity finite‑element model of the piping system is used to generate 8 × 10⁴ response simulations, establishing a reference fragility curve. The sequential DoE is then applied with sample sizes ranging from 30 to 80. Results show that the Bayesian estimates quickly converge to the reference curve, with credible intervals that remain tight even in regions where no failures are observed. The approach successfully avoids likelihood degeneracy, outperforms MLE‑based estimators, and achieves accurate parameter recovery with fewer than 100 data points.

In summary, the paper makes three key contributions: (1) a constrained Jeffreys’ prior that ensures a proper posterior for the probit‑lognormal fragility model under sparse binary data, (2) an information‑theoretic sequential DoE that selects the most informative seismic intensity levels, and (3) practical stopping criteria for efficient experimentation. The proposed framework is broadly applicable to any structural reliability problem where only binary outcomes are available, offering a cost‑effective path to robust fragility assessment in seismic risk analysis.