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

TL;DR: 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.

Summary

1 Introduction

• Each variable in a BN is associated with conditional distributions for the variable, one for each state of its parents.
• MTE functions are piecewise functions that are defined on regions called hypercubes, and the functions themselves are exponential functions of a linear function of the variables.
• An advantage of the MOP method is that one can easily find MOP approximations of differentiable PDFs using the Taylor series expansion of the PDF [8], or by using Lagrange interpolating polynomials [6].
• For dimensions two or greater, the hyper-rhombus condition is a generalization of the hypercube condition.
• Second, MOP functions defined on hyper-rhombuses are closed under operations required for multidimensional linear deterministic functions.

2.1 MOP Functions

• The definition given in Equation (1) is exactly the same as in Shenoy and West [8].
• The main motivation for defining MOP functions is that such functions are easy to integrate in closed form, and that they are closed under multiplication, integration, and addition, the main operations in making inferences in hybrid Bayesian networks.
• The definition of a m-dimensional MOP function stated in Equation (3) is more general than the corresponding definition stated in Shenoy and West [8], which is as follows:.
• It is easy to see that an m-dimensional function satisfying the condition in Equation (5) will also satisfy the condition in Equation (3), but the converse is not true.
• An advantage is that the authors can more easily construct high dimensional conditional PDFs such as the conditional linear Gaussian distributions.

3 Fitting MOPs to Two- and Three-Dimensional CLG PDFs

• The authors will find MOP approximations of the PDFs of 2- and 3- dimensional conditional linear Gaussian (CLG) distributions based on a MOP approximation of the 1-dimensional standard normal PDF.
• The authors revised definition of multi-dimensional MOP functions in Equation (3) facilitates the task of finding MOP approximations of the PDFs of CLG conditional distributions.

3.4 Three-Dimensional CLG Distributions

• As in the two-dimensional case, the authors will investigate how much of a time penalty one has to pay for using hyper-rhombus condition.
• The inner integral (with respect to y) required approximately 93 seconds (≈ 1.6 minutes), and resulted in a 2-dimensional, 7-degree, MOP.
• Thus, the two multiplications and the two integrations in Equation (17) require a total of approximately 269 seconds (or ≈ 4.5 minutes) using Mathematica c© on a laptop computer.
• In summary, the hyper-rhombus condition enables us to easily represent CLG conditionals in high dimensions.

4 Summary and Discussion

• A major contribution of this paper is a re-definition of multi-dimensional mixture of polynomials so that the regions where the polynomials are defined are hyperrhombuses instead of hypercubes.
• This re-definition allows us to use the MOP approximation of a one-dimensional standard normal PDF to define MOP approximations of high-dimensional CLG PDFs.
• Shenoy [6] compares the practical implications of the hyper-rhombus condition with the hypercube condition.
• He compares the time required for computation of marginals for a couple of simple Bayesian networks, and also the accuracy of the computed marginals.
• This is a topic that needs further investigation.

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

Fig. 3. A graph of h2(y) (in red) overlaid on the graph of fY (y) (in blue)

Fig. 5. A graph of h3(w) (in red) overlaid on the graph of fW (w) (in blue)

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.