Inference in Hybrid Bayesian Networks with Nonlinear Deterministic Conditionals

To enable inference in hybrid Bayesian networks containing nonlinear deterministic conditional distributions using mixtures of polynomials or mixtures of truncated exponentials, Cobb and Shenoy in 2005 propose approximating nonlinear deterministic functions by piecewise linear ones. In this paper, we describe a method for finding piecewise linear approximations of nonlinear functions based on two basic principles and an AIC-like heuristic. We illustrate our method for some commonly used onedimensional and two-dimensional nonlinear deterministic functions such as W = X2, W = eX , W = X · Y , and W = X/Y . Finally, we solve two small examples of hybrid Bayesian networks containing nonlinear deterministic conditionals that arise in practice.