Two issues in using mixtures of polynomials for inference in hybrid Bayesian networks

We discuss two issues in using mixtures of polynomials (MOPs) for inference in hy-brid Bayesian networks. MOPs were proposed by Shenoy and West for mitigating theproblem of integration in inference in hybrid Bayesian networks. First, in definingMOP for multi-dimensional functions, one requirement is that the pieces where thepolynomials are defined are hypercubes. In this paper, we discuss relaxing this condi-tion so that each piece is defined on regions called hyper-rhombuses. This relaxationmeans that MOPs are closed under transformations required for multi-dimensional lin-ear deterministic conditionals, such as Z = X + Y , etc. Also, this relaxation allows usto construct MOP approximations of the probability density functions (PDFs) of themulti-dimensional conditional linear Gaussian distributions using a MOP approxima-tion of the PDF of the univariate standard normal distribution. Second, Shenoy andWest suggest using the Taylor series expansion of differentiable functions for findingMOP approximations of PDFs. In this paper, we describe a new method for findingMOP approximations based on Lagrange interpolating polynomials (LIP) with Cheby-shev points. We describe how the LIP method can be used to find efficient MOPapproximations of PDFs. We illustrate our methods using conditional linear GaussianPDFs in one, two, and three dimensions, and conditional log-normal PDFs in one andtwo dimensions. We compare the efficiencies of the hyper-rhombus condition with thehypercube condition. Also, we compare the LIP method with the Taylor series method.