On Undirected Representations of Bayesian Networks

Empirical studies clearly demonstrate the effectiveness of the nested jointree (NJT) representation in probabilistic inference. A NJT is a traditional Markov network (MN) together with a possible local MN nested in each clique. These nested MNs can themselves contain other nested MNs in a recursive manner. However, the NJT representation is not necessarily a faithful representation of a given Bayesian network (BN). This means that the effectiveness of a NJT has been demonstrated while only exploiting some of the independency information available in the given BN. In this paper, we introduce a new kind of probabilistic network, called a hierarchical Markov network (HMN). A HMN is a hierarchy of MNs. We give an algorithm to transform a BN into a canonical HMN. The main result of this paper is that the constructed HMN is unique and equivalent to the input BN. Since HMNs are a faithful representation of BNs, a query may be optimized using independencies in a HMN that otherwise would have gone unrepresented in a NJT approach.