Seismic risk analysis using Bayesian belief networks - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 174

is quantifi ed by conditional probabilities for each node given its parents in
the network, or conditional probability tables (CPTs); each variable is
assigned a CPT of the variable given its parents. For variables with no
parents, the probabilities are reduced to the unconditional probability (UP)
(e.g. A 1 and A 2 in Fig. 7.1). The effi cacy of a BBN is realized in its fl exibility
to capture top-down inference, observing the cause (or parent) and infer-
ring the possible effect (or child), and bottom-up inference, observing the
effect (child) and inferring the possible cause (parent).
The main concept of a BBN is rooted in the use of Bayes' theorem, in
which the relation between two nodes, hypothesis H (parent) and evidence
E (child), is represented as:

( ) × ()

()

pEH pH
pE

( ) = PHE [7.1]

where p ( H | E ) is one's belief for hypothesis H upon observing evidence E ,
p ( E | H ) is the likelihood that E is observed if H is true, p ( H ) is the probabil-
ity that the hypothesis holds true, and p ( E ) is the probability that the evi-
dence takes place. p ( H | E ) is known as posterior probability and p ( H ) is
called prior probability. In a BBN analysis, for n number of mutually exclu-
sive hypotheses H i ( i =1, . . . , n ) , and a given evidence E , the updated probability
is computed by expanding the p ( E ) in Equation (7.1) as:

( ) × ()

pEH pH

( ) =

j j

pH E [7.2]

j

n

∑

( ) × ()

pEH pH

i i

i = 1

Fundamentally, a BBN is used to update probabilities as new information
is obtained. The network supports the computation of the probabilities of
any subset of variables given evidence about any other subset.

7.2.1 Conditional probabilities

The conditional probabilities shown in Fig. 7.1, which will be used in Equa-
tions 7.1 and 7.2, can be obtained through expert knowledge elicitation
(Katsis et al. 2008; Joseph et al. 2010), or training from data (Cooper and
Herskovits 1992; Bouckaert et al. 2011). Where multiple experts are consid-
ered, credibility of each decision maker on the decision can be elicited by
considering experience and confi dence on the assessment (Kiremidjian
1985; Tesfamariam et al. 2010). The credibility factor can be used to reach
the overall decision.
The UPs of the basic input parameters are often not known a priori .
Consequently, equal weights (1/ n , where n is number of category considered
for each basic input) can be assigned using the principle of insuffi cient
reasoning (Tesfamariam and Martín-Pérez 2008). For example, if the states