Being Bayesian about Network Structure

In many domains, we are interested in analyzing the structure of the underlying distribution, e.g., whether one variable is a direct parent of the other. Bayesian model-selection attempts to find the MAP model and use its structure to answer these questions. However, when the amount of available data is modest, there might be many models that have non-negligible pos-terior. Thus, we want compute the Bayesian poste-rior of a feature, i.e., the total posterior probability of all models that contain it. In this paper, we propose a new approach for this task. We first show how to ef-ficiently compute a sum over the exponential number of networks that are consistent with a fixed ordering over network variables. This allows us to compute, for a given ordering, both the marginal probability of the data and the posterior of a feature. We then use this result as the basis for an algorithm that approx-imates the Bayesian posterior of a feature. Our ap-proach uses an Markov Chain Monte Carlo (MCMC) method, but over orderings rather than over network structures. The space of orderings is much smaller and more regular than the space of structures, and has a smoother posterior “landscape”. We present empiri-cal results on synthetic and real-life datasets that com-pare our approach to full model averaging (when pos-sible), to MCMC over network structures, and to a non-Bayesian bootstrap approach.