Why BDeu? Regular Bayesian network structure learning with discrete and continuous variables

We consider the problem of Bayesian network structure learning (BNSL) from data. In particular, we focus on the score‐based approach rather than the constraint‐based approach and address what score we should use for the purpose. The Bayesian Dirichlet equivalent uniform (BDeu) has been mainly used within the community of BNs (not outside of it). We know that for any model selection and any data, the fitter the data to a model, the more complex the model, and vice versa. However, recently, it was proven that BDeu violates regularity, which means that it does not balance the two factors, although it works satisfactorily (consistently) when the sample size is infinitely large. In addition, we claim that the merit of using the regular scores over the BDeu is that tighter bounds of pruning rules are available when we consider efficient BNSL. Finally, using experiments, we compare the performances of the procedures to examine the claim. (This paper is for review and gives a unified viewpoint from the recent progress on the topic.)