Triangulation by Continuous Embedding

Belief networks are graphical representations of probability distributions over a set of variables. In what follows it will be always assumed that the variables take values in a finite set and that they correspond to the vertices of a graph. The graph's arcs will represent the dependencies among variables. There are two kinds of representations that have gained wide use: one is the directed acyclic graph model, also called a Bayes net, which represents the joint distribution as a product of the probabilities of each vertex conditioned on the values of its parents; the other is the undirected graph model, also called a Markov field, where the joint distribution is factorized over the cliques! of an undirected graph. This factorization is called a junction tree and optimizing it is the subject of the present paper. The power of both models lies in their ability to display and exploit existent marginal and conditional independencies among subsets of variables.