On Bayesian Dirichlet Scores for Staged Trees and Chain Event Graphs

Chain event graphs (CEGs) are a recent family of probabilistic graphical models that have emerged as a suitable alternative to Bayesian networks (BNs) for asymmetric processes. These asymmetries include context-specific independencies and structural asymmetries (i.e. structural zeros and structural missing values). Model selection in CEGs is done through an intermediate model called staged trees, and similar to BNs, this can be done through a score-based approach. Moreover, a CEG is uniquely defined by its staged tree. In BNs, the Bayesian Dirichlet equivalent uniform (BDeu) score - obtained through a specific hyperparameter setting in the Bayesian Dirichlet score function - is popular for score-based model selection for its desirable theoretical properties such as ease of hyperparameter setting, preservation of effective sample size, and score equivalence. It has been shown that, under standard assumptions, the BD score function can analogously be defined for staged trees and thereby, for CEGs. However, unlike in BNs, there has been little research into the effects of hyperparameter setting in the BD score function on both these models. In this paper, we derive a BDeu score for staged trees and CEGs. Further, we explore the relationship between the BD sparse (BDs) score, proposed for BNs that contain unobserved configurations of its variables within a dataset, and the BDeu for staged trees and CEGs. Through this relationship, we demonstrate the favourable properties of CEGs in modelling processes with sparsity or asymmetry.