Risk Assessment and Decision Analysis with Bayesian Networks

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6

From Bayes’ Theorem to

Bayesian Networks

6.1 Introduction

We have now seen how Bayes’ theorem enables us to correctly update a

prior probability for some unknown event when we see evidence about the

event. But in any real-world risk assessment problem there will be many

unknown events and many different pieces of evidence, some of which

may be related. We already saw in Chapter 2 examples of such risk assess-

ment problems. When we represent such problems graphically (with all

uncertain variables being represented as nodes and an edge between two

nodes representing a relationship) we have a Bayesian network (BN). It

turns out that Bayes’ theorem can be applied to correctly update evidence

in these more general complex problems, and the purpose of this chapter

is to explain exactly how this is done and under what constraints.

We start in Section 6.2 by considering the simplest type of risk

assessment problem with just two variables. We can think of this as a

two-node BN. In this case if we enter evidence on one variable we can

use Bayes’ theorem exactly as described in Chapter 5 to update the

probability of the other variable. So essentially Section 6.2 explains

fully how to do computations in a two-node BN. In Section 6.3 we

show how Bayes’ theorem can be extended to perform the necessary

probability updating when we consider an extended version of the

problem (with four nodes). The calculations quickly become quite

tricky, but in Section6.4 we demonstrate how the underlying idea of

Bayesian “propagation” through the network provides some very pow-

erful types of reasoning. This leads to a formal denition of BNs in

Section 6.5, which is based on the crucial notion of what is meant

when two variables are directly linked (the so-called independence

assumptions). In Section 6.6 we describe the basic structural proper-

ties of BNs, and it is a recognition of these properties that form the

basis for the general propagation algorithm that is described in

Section6.7. In Section 6.8 we show how BNs can be used to “solve”

two of the paradoxes discussed in earlier chapters (Monty Hall and

Simpson). Finally, in Section 6.9 we provide guidelines for building

and running BN models.

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

this chapter

Readers who do not wish to under-

stand any of the mathematical

underpinnings of BNs could go

straight to Section 6.9 for a practi-

cal description of how to build and

run BNs.