Hierarchical Bayesian models with subdomain clustering for parameter estimation of discrete Bayesian network

Bayesian network (BN) is a powerful tool for the probabilistic modeling and inference of multiple random variables. While conditional probability tables (CPTs) of a discrete BN provide a unified representation facilitating closed-form inference by efficient algorithms, they pose challenges in parameter estimation, especially due to data sparsity resulting from the discretization of continuous parent variables. To address the challenges, this paper presents a novel BN modeling approach, which is the first attempt to apply hierarchical Bayesian modeling to quantify the CPT of a child variable with discretized multiple parent variables. In addition, given that discretization results in many subdomains showing strong correlation, the concept of subdomain clustering is introduced in both supervised and unsupervised learning schemes. The proposed procedure is demonstrated by its application to the BN model describing structural responses under a sequence of main and aftershocks. In the model, the structural dynamic response of interest is modeled by a CPT in discretized domains of six-dimensional ground motion features. Hierarchical Bayesian normal models are developed to quantify the conditional probability parameters in the subdomains, which are classified using the information of peak ground acceleration. The proposed approach facilitates robust parameter estimation of the CPT, especially in the subdomains with a small number of data points. This is thoroughly validated by comparing the inference results of the CPT by the proposed method with those by an alternative approach that does not consider the correlation between subdomains. Furthermore, the validation is performed on different subsets of the parent variables with various unsupervised learning schemes to demonstrate the general effectiveness of the subdomain clustering for the hierarchical Bayesian approach.