Abstract
Structure learning is one of the main concerns in studies of Bayesian networks. In the present paper, we consider networks consisting of both observable and hidden nodes, and propose a method to investigate the existence of a hidden node between observable nodes, where all nodes are discrete. This corresponds to 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 structure learning based on the Laplace approximation do not work. The proposed method is based on 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.
Highlights
In learning Bayesian networks, one of the main concerns is structure learning
Many criteria to detect the network structure have been proposed such as the minimum description length (MDL) [1], the Bayesian information criterion (BIC) [2], the Akaike information criterion (AIC) [3], and the marginal likelihood [4]
Based on the relation p( Z n | X n, Y n ) ∝ p( X n, Z n, Y n ), the Markov Chain Monte Carlo (MCMC) method provides the sampling of Z n from p( Z n | X n, Y n )
Summary
Many criteria to detect the network structure have been proposed such as the minimum description length (MDL) [1], the Bayesian information criterion (BIC) [2], the Akaike information criterion (AIC) [3], and the marginal likelihood [4] Most of these criteria assume statistical regularity, which means that the network has identifiability on the parameter and the nodes are observable. We consider a two-step method; the first step obtains the optimal structure with observable nodes and the second step detects the hidden nodes in each partial graph.
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