STRUCTURE OF THE HIGH-ORDER BOLTZMANN MACHINE FROM INDEPENDENCE MAPS

In this paper we consider the determination of the structure of the hi gh-order Boltzmann machine (HOBM), a stochastic recurrent network for approximating probability distributions, We obtain the structure of th e HOBM, the hypergraph of connections, from conditional independences of the probability distribution to model, We assume that an expert pro vides these conditional independences and from them we build independe nce maps, Markov and Bayesian networks, which represent conditional in dependences through undirected graphs and directed acyclic graphs resp ectively, From these independence maps we construct the HOBM hypergrap h, The central aim of this paper is to obtain a minimal hypergraph, Gi ven that different orderings of the variables provide in general diffe rent Bayesian networks, we define their intersection hypergraph, We pr ove that the intersection hypergraph of all the Bayesian networks (N!) of the distribution is contained by the hypergraph of the Markov netw ork, it is more simple, and we give a procedure to determine a subset of the Bayesian networks that verifies this property, We also prove th at the Markov network graph establishes a minimum connectivity for the hypergraphs from Bayesian networks.