Empirical Bayes Approach for Developing Hierarchical Probabilistic Predictive Models and Its Application to the Seismic Reliability Analysis of FRP-Retrofitted RC Bridges

Abstract

This paper proposes a general formulation to develop hierarchical probabilistic predictive models for clustered data. The common clustering factor, shared among the data within a group, causes statistical dependence that needs to be accounted for in the estimation of unknown model parameters. The basic idea of the hierarchical formulation is that the unknown model parameters are endowed with distributions that depend on a set of shared underlying parameters, and this construction is recursive up to the highest level of the hierarchy. The usual improper noninformative prior distributions on variance parameters of hierarchical models can lead to nonexistent posterior distributions that may appear perfectly reasonable in numerical simulations. On the other hand, common proper noninformative prior distributions may also substantially affect posterior statistics. Instead, the empirical Bayes approach is proposed to objectively estimate the variance parameters. The Gibbs sampling algorithm is used to estimate the unknown model parameters in the context of a Bayesian updating approach. The proposed formulation is used to develop probabilistic seismic deformation demand models for reinforced concrete (RC) bridges retrofitted with fiber-reinforced polymer (FRP) composites. The developed demand models are then used with previously developed probabilistic deformation capacity models to objectively assess the reduction in conditional failure probability of bridges due to the FRP retrofitting for given ground motion intensities. Furthermore, two formulations to compute the unconditional annual failure probability are also developed along with a closed-form solution of one of them. The proposed formulations are illustrated considering three example RC bridges.