A re-definition of mixtures of polynomials for inference in hybrid Bayesian networks

TLDR: This relaxation means that MOPs are closed under transformations required for multi-dimensional linear deterministic conditionals, such as Z = X + Y, which allows us to construct MOP approximations of the probability density functions (PDFs) of theMulti-dimensional conditional linear Gaussian distributions using a MOP approximation of the PDF of the univariate standard normal distribution.

Abstract: We discuss some issues in using mixtures of polynomials (MOPs) for inference in hybrid Bayesian networks. MOPs were proposed by Shenoy and West for mitigating the problem of integration in inference in hybrid Bayesian networks. In definingMOP for multi-dimensional functions, one requirement is that the pieces where the polynomials are defined are hypercubes. In this paper, we discuss relaxing this condition so that each piece is defined on regions called hyper-rhombuses. This relaxation means that MOPs are closed under transformations required for multi-dimensional linear deterministic conditionals, such as Z = X + Y. Also, this relaxation allows us to construct MOP approximations of the probability density functions (PDFs) of the multi-dimensional conditional linear Gaussian distributions using a MOP approximation of the PDF of the univariate standard normal distribution. We illustrate our method using conditional linear Gaussian PDFs in two and three dimensions.

Figures

Fig. 1. A graph of g1(z) versus z (in red) overlaid on the graph of ϕ(z) (in blue)

Fig. 8. A graph of h8(v) (in red) overlaid on the graph of fV (v) (in blue)

Fig. 2. A 3-dimensional plot of h1(z, y)

Fig. 6. A graph of h6(x) (in red) overlaid on the graph of fX(x) (in blue)

Fig. 7. A Bayesian network with a 3-dimensional conditional

Fig. 4. A Bayesian network with a sum deterministic function

Frequently asked questions

Q1. What have the authors contributed in "A re-definition of mixtures of polynomials for inference in hybrid bayesian networks" ?

The authors discuss some issues in using mixtures of polynomials ( MOPs ) for inference in hybrid Bayesian networks. In this paper, the authors discuss relaxing this condition so that each piece is defined on regions called hyper-rhombuses.