Stochastic independence is an idealized relationship located at one end of a continuum of values measuring degrees of dependence. Modeling real world systems, we are often not interested in the distinction between exact independence and any degree of dependence, but between weak ignorable and strong substantial dependence. Good models map significant deviance from independence and neglect approximate independence or dependence weaker than a noise threshold. This intuition is applied to learning the structure of Bayes nets from data. We determine the conditional posterior probabilities of structures given that the degree of dependence at each of their nodes exceeds a critical noise level. Deviance from independence is measured by mutual information. Arc probabilities are determined by the amount of mutual information the neighbors contribute to a node, is greater than a critical minimum deviance from independence. A Ø 2 approximation for the probability density function of mutual info...

When a Bayes net is learned from data, the arc structure is inferred probabilistically only. It is of central interest, thus, to have a formal system that provides the methods to treat the probability of such arc structures. The present contribution introduces a probability function on the family of sets of directed acyclic graphs so that the assignment of probabilities to arc structures has a well defined semantics.

Random graphs are used to model the qualitative structure of Bayesian networks. The arc