FUZZY BAYESIAN NETWORKS — A GENERAL FORMALISM FOR REPRESENTATION, INFERENCE AND LEARNING WITH HYBRID BAYESIAN NETWORKS

Abstract

This paper proposes a general formalism for representation, inference and learning with general hybrid Bayesian networks in which continuous and discrete variables may appear anywhere in a directed acyclic graph. The formalism fuzzifies a hybrid Bayesian network into two alternative forms: the first form replaces each continuous variable in the given directed acyclic graph (DAG) by a partner discrete variable and adds a directed link from the partner discrete variable to the continuous one. The mapping between two variables is not crisp quantization but is approximated (fuzzified) by a conditional Gaussian (CG) distribution. The CC model is equivalent to a fuzzy set but no fuzzy logic formalism is employed. The conditional distribution of a discrete variable given its discrete parents is still assumed to be multinomial as in discrete Bayesian networks. The second form only replaces each continuous variable whose descendants include discrete variables by a partner discrete variable and adds a directed link from that partner discrete variable to the continuous one. The dependence between the partner discrete variable and the original continuous variable is approximated by a CG distribution, but the dependence between a continuous variable and its continuous and discrete parents is approximated by a conditional Gaussian regression (CGR) distribution. Obviously, the second form is a finer approximation, but restricted to CGR models, and requires more complicated inference and learning algorithms. This results in two general approximate representations of a general hybrid Bayesian networks, which are called here the fuzzy Bayesian network (FBN) form-I and form-II. For the two forms of FBN, general exact inference algorithms exists, which are extensions of the junction tree inference algorithm for discrete Bayesian networks. Learning fuzzy Bayesian networks from data is different from learning purely discrete Bayesian networks because not only all the newly converted discrete variables are latent in the data, but also the number of discrete states for each of these variables and the CG or CGR distribution of each continuous variable given its partner discrete parents or both continuous and discrete parents have to be determined.