  One of the key elements of probabilistic seismic risk assessment studies is
the fragility curve, which represents the conditional probability of failure of
a mechanical structure for a given scalar measure derived from seismic ground
motion. For many structures of interest, estimating these curves is a daunting
task because of the limited amount of data available; data which is only binary
in our framework, i.e., only describing the structure as being in a failure or
non-failure state. A large number of methods described in the literature tackle
this challenging framework through parametric log-normal models. Bayesian
approaches, on the other hand, allow model parameters to be learned more
efficiently. However, the impact of the choice of the prior distribution on the
posterior distribution cannot be readily neglected and, consequently, neither
can its impact on any resulting estimation. This paper proposes a comprehensive
study of this parametric Bayesian estimation problem for limited and binary
data. Using the reference prior theory as a cornerstone, this study develops an
objective approach to choosing the prior. This approach leads to the Jeffreys
prior, which is derived for this problem for the first time. The posterior
distribution is proven to be proper with the Jeffreys prior but improper with
some traditional priors found in the literature. With the Jeffreys prior, the
posterior distribution is also shown to vanish at the boundaries of the
parameters' domain, which means that sampling the posterior distribution of the
parameters does not result in anomalously small or large values. Therefore, the
use of the Jeffreys prior does not result in degenerate fragility curves such
as unit-step functions, and leads to more robust credibility intervals. The
numerical results obtained from different case studies-including an industrial
example-illustrate the theoretical predictions.
