Gaussian Bayesian network for reliability analysis of a system of bridges

A Gaussian Bayesian Network (GBN) is a special directed graphical model with conditional Gaussian distributions. It is an efficient statistical tool in the development of decision support systems, because it offers exact algorithms for prediction and inference. Properly, a GBN requires all variables to be defined by a Gaussian prior distribution or by a Gaussian conditional distribution, whose mean is linearly related to the parent variables and whose variance is constant. This paper applies GBN to the management of a network of bridges after a seismic event. Knowledge about the experienced damage is gained from estimation of the magnitude and epicenter location and, thereby, the intensity of the ground motion and from observations collected in the field through visual inspections and sensor recordings of the structural response. Certain observations, such as the observed damage state of a bridge, cannot be described by a Gaussian likelihood function. For example observing that a bridge has collapsed is equivalent to observing that the bridge capacity is lower than the seismic demand. To include this type of evidence in the model, we adopt numerical schemes based on importance sampling and Gibbs sampling. The proposed method is illustrated through its application to the reliability assessment of a large bridge network. alytical and approximate methods for making inference with non-Gaussian likelihood functions. Section 6 presents a numerical application and Section 7 draws conclusions. 2 INFERENCE IN BAYESIAN NETWORKS WITH GAUSSIAN VARIABLES In this section, we recap the basic assumptions for a GBN, as presented in Pozzi et al. (2012). In a GBN, the joint probability of all variables is Gaussian, and consequently each marginal or conditional is Gaussian as well. If vector is a root in the BN graph, we require the joint distribution of to be Gaussian. If it is a child, we require that its conditional distribution be of the form: