Modeling Deterministic Equations in Discrete Bayesian Networks by Credal Networks

Abstract

We focus on the problem of embedding deterministic equations in discrete Bayesian networks. This is typically achieved by a discretization of both input and output variables and a degenerate quantification of the corresponding conditional probability table. We note that, generally speaking, this approach based on the classical Bayesian theory of probability cannot properly take into account the information loss induced by the discretization. Consequently, we propose a conservative modeling of such epistemic uncertainty by means of credal sets, i.e., sets of probability mass functions. This approach transforms the original Bayesian network in a credal network, returning interval-valued inferences, that are robust with respect to the discretization. Procedures for optimal discretization in this setup are discussed. The proposed method can be used for both developing knowledge-based expert systems as well as machine learning based on probabilistic graphical models.