Risk Assessment and Decision Analysis with Bayesian Networks

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9

Numeric Variables and Continuous

Distribution Functions

9.1 Introduction

In most real-world risk assessment applications we cannot rely on all

variables of interest being of the types covered in Chapter 8, that is, either

labeled, Boolean, or ranked. Inevitably, we will need to include variables

that are numeric. Such variables could be discrete (such as counts of the

number of defects in a product) or continuous (such as the water level in

centimeters in a river); they generally require an innite number of states.

In Section 9.2 we show that much of the theory, notation, and ideas

encountered in earlier chapters apply equally well to numeric nodes.

This section also provides a useful introduction into some of the special-

ist terminology used by Bayesian statisticians, who mainly use numeri-

cal variables in their inference models.

A major advantage of dening a node as a numeric node (as opposed to,

say, a labeled or even ranked node) is that we are able to use a wide range

of pre-dened mathematical and statistical functions instead of having to

manually dene node probability tables (NPTs). But there is also a problem,

which until recently was truly the Achilles heel of BNs. The standard (exact)

inference algorithms for Bayesian network (BN) propagation (as described

in Chapter 6, and Appendices B and C) only work in the case where every

node of the BN has a nite set of discrete states. Although it is always pos-

sible to map a numeric node into a pre-dened nite set of discrete states

(a process called static discretization), the result of doing so is an inevitable

loss of accuracy. We will explain why in Section 9.3. In most real-world

applications this loss of accuracy makes the model effectively unusable.

Moreover, as we shall see, increasing the granularity of static discretization

(at the expense of model efciency) rarely provides a workable solution.

Fortunately, recent breakthroughs in algorithms for BNs that incor-

porate both numeric and nonnumeric nodes (called hybrid BNs) have

provided a solution to the problem using an algorithm called dynamic

discretization, which works efciently for a large class of continuous

distributions. This is explained in Section 9.4. Users of AgenaRisk,

which implements this algorithm, do not have to worry much about pre-

dening the set of states of any numeric node.

Visit www.bayesianrisk.com for your free Bayesian network software and models in

this chapter

So far in this book we have dened

discrete probability distributions for

discrete nodes. In this chapter we

will dene continuous probability

distributions for numerical nodes.

The ideas and concepts intro-

duced in Chapters 4, 5, and 6 still

apply but there are some additional

mathematical concepts that will be

needed to support these numeric

nodes properly.

AgenaRisk denes the use of the

dynamic discretization algorithm

as “simulation”—all approximate

methods for solving this problem

are dubbed simulation because they

“simulate” or approximate the true

mathematical function. Details of

the dynamic discretization algo-

rithm are given in Appendix D.