Risk Assessment and Decision Analysis with Bayesian Networks

215

8

Building and Eliciting Node

Probability Tables

8.1 Introduction

This chapter discusses one of the main challenges encountered when

building a Bayesian network (BN) and attempting to complete the node

probability tables (NPTs) in the BN. This is that the number of proba-

bility values needed from experts can be unfeasibly large, despite the

best efforts to structure the model properly (Section8.2). We attempt

to address this problem by introducing methods to complete NPTs that

avoid us having to dene all the entries manually. Specically, we use

functions of varying sorts and give guidance on which functions work

best in different situations. The particular functions and operators that

we can use depend on the type of node: In this chapter we focus on

discrete node types: labeled, Boolean, discrete real, and ranked. In

Section8.3 we show that for labeled nodes comparative expressions

(e.g., IF, THEN, ELSE) can be used, and in Section 8.4 for Boolean

nodes we show that we can also use a range of Boolean functions

including OR, AND, and a special function called NoisyOR. We also

show how to implement the notion of a weighted average. In Section

8.5 we describe a range of functions that can be used for ranked nodes.

Finally Section 8.6 describes the challenges of eliciting from experts

probabilities and functions that generate NPTs. This draws on practical

lessons and knowledge of cognitive psychology.

8.2 Factorial Growth in the Size

of Probability Tables

Recall that the NPT for any node of a BN (except for nodes with-

out parents) is intended to capture the strength of the relationship

between the node and its parents. This means we have to dene

the probability of the node, given every possible state of the parent

nodes. In principle this looks like a daunting problem as explained

in Box8.1.

The previous two chapters pro-

vided concrete guidelines on how

to build the structure, that is, the

graphical part, of a BN model.

Although this is a crucial rst step,

we still have to add the node prob-

ability tables (NPTs) before the

model can be used.

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

this chapter

The focus of this chapter is on dis-

crete variables. We will deal with

probabilities for continuous vari-

ables in Chapter 9.

216 Risk Assessment and Decision Analysis with Bayesian Networks

Box8.1 Factorial Growth in Probability Tables

Consider a simple BN shown in Figure8.1. The NPT for the node Water level at end of day has 3 states and its

parent nodes have 3 and 4 states, respectively, making 36 possible state combinations to consider. When you

create this BN model in a tool like AgenaRisk and examine the NPT for the node Water level at end of day you

will see a table with all cells equal:

Water

Level at

Start of

Day Low Medium High

Rainfall None

<2

mm

2–5

mm

>5

mm None

<2

mm

2–5

mm

>5

mm None

<2

mm

2–5

mm

>5

mm

Low 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Medium 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

High 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Your task is enter meaningful probability values so that it looks something like this:

Water

Level at

Start of

Day Low Medium High

Rainfall None

<2

mm

2–5

mm

>5

mm None

<2

mm

2–5

mm

>5

mm None

<2

mm

2–5

mm

>5

mm

Low 1.0 0.7 0.1 0.0 0.1 0.0 0.0 0.0 0.1 0.0 0.0 0.0

Medium 0.0 0.3 0.5 0.2 0.9 0.8 0.2 0.1 0.4 0.1 0.0 0.0

High 0.0 0.0 0.4 0.8 0.0 0.2 0.8 0.9 0.5 0.9 1.0 1.0

Although a 36-cell table might be manageable, suppose that we decided to change the states of the water

level nodes to be: very low, low, medium, high, very high. Then the NPT now has 5 × 5× 4 = 100 cells. If we

add another parent with 5 states then this jumps to 500 entries, and if we increase the granularity of rainfall to,

say, 10 states then the NPT has 1,250 cells. There is clearly a problem of combinatorial explosion.

Water level at

start of day

Water level at

end of day

Rainfall

States:

low, medium,

high

States:

low, medium,

high

States:

None,

< 2 mm,

2–5 mm

,

>5 mm

Figure 8.1 A simple BN.