Finding Optimal Bayesian Networks

In this paper, we derive optimality results for greedy Bayesian-network search algorithms that perform single-edge modifications at each step and use asymptotically consistent scoring criteria. Our results extend those of Meek (1997) and Chickering (2002), who demonstrate that in the limit of large datasets, if the generative distribution is perfect with respect to a DAG defined over the observable variables, such search algorithms will identify this optimal (i.e. We relax their assumption about the generative distribution, and assume only that this distribution satisfies the composition property over the observable variables, which is a more realistic assumption for real domains. Under this assumption, we guarantee that the search algorithms identify an inclusion-optimal model; that is, a model that (1) contains the generative distribution and (2) has no sub-model that contains this distribution. In addition, we show that the composition property is guaranteed to hold whenever the dependence relationships in the generative distribution can be characterized by paths between singleton elements in some generative graphical model (e.g. a DAG, a chain graph, or a Markov network) even when the generative model includes unobserved variables, and even when the observed data is subject to selection bias. Introduction The problem of learning Bayesian networks (a.k.a directed graphical models) from data has received much attention in the UAI community. A simple approach taken by many researchers, particularly those contributing experimental papers, is to apply--in conjunction with a scoring criterion--a greedy single-edge search algorithm to the space of Bayesian-network structures or to the space of equivalence classes of those structures. There are a number of important reasons for the popularity of this approach.