Asymptotic Model Selection for Directed Networks with Hidden Variables

Abstract

We extend the Bayesian Information Criterion (BIC), an asymptoticapproximation for the marginal likelihood, to Bayesian networks withhidden variables. This approximation can be used to select modelsgiven large samples of data. The standard BIC as well as our extensionpunishes the complexity of a model according to the dimension ofits parameters. We argue that the dimension of a Bayesian networkwith hidden variables is the rank of the Jacobian matrix of the transformationbetween the parameters of the network and the parameters of the observablevariables. We compute the dimensions of several networks includingthe naive Bayes model with a hidden root node.