Fx. Albizuri et al., STRUCTURE OF THE HIGH-ORDER BOLTZMANN MACHINE FROM INDEPENDENCE MAPS, IEEE transactions on neural networks, 8(6), 1997, pp. 1351-1358
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.