Structure of the high-order Boltzmann machine from independence maps

Abstract

In this paper we consider the determination of the structure of the high-order Boltzmann machine (HOBM), a stochastic recurrent network for approximating probability distributions. We obtain the structure of the HOBM, the hypergraph of connections, from conditional independences of the probability distribution to model. We assume that an expert provides these conditional independences and from them we build independence maps, Markov and Bayesian networks, which represent conditional independences through undirected graphs and directed acyclic graphs respectively. From these independence maps we construct the HOBM hypergraph. The central aim of this paper is to obtain a minimal hypergraph. Given that different orderings of the variables provide in general different Bayesian networks, we define their intersection hypergraph. We prove that the intersection hypergraph of all the Bayesian networks (N!) of the distribution is contained by the hypergraph of the Markov network, it is more simple, and we give a procedure to determine a subset of the Bayesian networks that verifies this property. We also prove that the Markov network graph establishes a minimum connectivity for the hypergraphs from Bayesian networks.

Singularities in Complete Bipartite Graph-Type Boltzmann Machines and Upper Bounds of Stochastic Complexities

“…In naive Bayesian networks, the relation between the stochastic complexities and the singularities has been clarified [14]. We can regard general Boltzmann machines as graphical models and Bayesian networks [2]. If we add the connections between hidden units to the models of Corollary 1, the models are equivalent to general Boltzmann machines.…”