Hidden Node Detection between Two Observable Nodes Based on Bayesian Clustering

The structure learning is one of the main concerns in studies of the Bayesian networks. In the present paper, we consider the network consisting of both observable and hidden nodes, and propose a method to investigate the existence of a hidden node between two observable nodes, which is the model selection problem between the networks with and without the middle hidden node. When the network includes a hidden node, it has been known that there are singularities in the parameter space, and the Fisher information matrix is not positive definite. Then, the many conventional criteria for the structure learning based on the Laplace approximation do not work. The proposed method is based on the Bayesian clustering, and its asymptotic property justifies the result; the redundant labels are eliminated and the simplest structure is detected even if there are singularities.