Efficient Approximations for the Marginal Likelihood of Bayesian Networks with Hidden Variables

We discuss Bayesian methods for model averaging and model selection among Bayesian-network models with hidden variables. In particular, we examine large-sample approximations for the marginal likelihood of naive-Bayes models in which the root node is hidden. Such models are useful for clustering or unsupervised learning. We consider a Laplace approximation and the less accurate but more computationally efficient approximation known as the Bayesian Information Criterion (BIC), which is equivalent to Rissanen's (1987) Minimum Description Length (MDL). Also, we consider approximations that ignore some off-diagonal elements of the observed information matrix and an approximation proposed by Cheeseman and Stutz (1995). We evaluate the accuracy of these approximations using a Monte-Carlo gold standard. In experiments with artificial and real examples, we find that (1) none of the approximations are accurate when used for model averaging, (2) all of the approximations, with the exception of BIC/MDL, are accurate for model selection, (3) among the accurate approximations, the Cheeseman–Stutz and Diagonal approximations are the most computationally efficient, (4) all of the approximations, with the exception of BIC/MDL, can be sensitive to the prior distribution over model parameters, and (5) the Cheeseman–Stutz approximation can be more accurate than the other approximations, including the Laplace approximation, in situations where the parameters in the maximum a posteriori configuration are near a boundary.