Essential Graphs and Bayesian Networks

Our purpose in this paper is to show how it is possible to obtain improved tools, in a computational sense, for Learning Bayesian Networks (LBNs). And also give a more mathematically consistent and complete formulation. First, by the partition in equivalence classes, and then selecting a graph as representative of each one of them, the so called Essential Graph. Second, analyzing the asymptotical behavior of the ratio among the cardinal of equivalence classes (therefore, of essential graphs) and the cardinal of Directed Acyclic Graphs (DAGs) of order n, and reciprocally, when this number of nodes tends to infinity. This study is made in both algebraic and geometrical ways. Finally, we describe the future research lines, in LBNs and Probabilistic Graphical Models (PGMs), by new algebraic and geometrical tools.