Bayesian Network Structure Estimation Based on the Bayesian/MDL Criteria When Both Discrete and Continuous Variables Are Present

We consider estimation of Bayesian network structures given a finite number of examples when both discrete and continuous random variables are present in a Bayesian network. It is not hard to estimate Bayesian network structures based on the MDL/Bayesian criteria if each variable takes a finite value. On the other hand, because continuous data contain infinite precisions, its posterior probability cannot be evaluated in a well defined manner. We extend the notion of the MDL/Bayesian criteria in the most general setting in terms of Radon-Nikodym derivatives, and propose a method to estimate Bayesian network structures without assuming each variable to be either discrete or continuous.