Learning Bayesian Networks

We examine Bayesian methods for learning Bayesian networks from a combination of prior knowledge and statistical data. In particular, we develop simple methods for generating priors for Bayesian-network parameters. Our work is a generalization of previous work that has concentrated on Bayesian networks containing only discrete variables and (to a lesser extent) on Gaussian networks. We introduce three assumptions that are abstractions of previously made assumptions: likelihood equivalence, which says that data should not help to discriminate network structures that represent the same assertions of conditional independence, parameter independence, which says that the parameters associated with each node in a Bayesian network are independent, and parameter modu-larity, which says that if a node has the same parents in two distinct networks, then the probability density functions of the parameters associated with this node are identical in both networks. We show how these assumptions greatly simplify the construction of priors. In addition, we use these assumptions to derive a general metric for complete databases. Combining this general metric with well-known statistical facts about the Dirichlet and normal{Wishart distribution, we provide simple derivations of metrics for discrete and Gaussian networks, respectively. Finally, we show how our assumptions lead to a general framework for characterizing prior distributions for the parameters of multivariate sampling.