that marginalization is always performed over the smallest possible number of product terms
the table dimensions required to manipulate may be efficiently reduced.
The considered example already has shown how evidence can be considered in the Bayesian
Networks. In the example only the type of evidence was considered where it is known that
one of the variables is in one particular state. However, evidence can be introduced formally
and in more general terms through the chain rule as applied in Equation (10.1).
If it is assumed that evidence e is available in terms of statements such as; “the variable A
with n possible states for some reason can only attain realizations in state i or j with
probabilities or ” then the joint probability distribution is given as
. It is seen that this probability distribution is achieved
simply through the multiplication of with the vector (. Such
vectors (or tables) are also denoted findings
ai
,.., 0aj
PAe( , ) (0, 0 , ,.aij0.,a,..0,0)
PA() 0, 0,.., 0,1, 0,..,1, 0,.., 0, 0)
e.In general terms there is:
PeP(,) ()UU e (10.5)Using the principle of Equation (10.5) on the chain rule as given in Equation (10.1) assuming
that evidence represented in terms of m findings is available there is:
1(,) ( ( ))
m
ii
ijPe PApaA e
U .. j (10.6)
Finally conditional probability distribution functions PA e( j ) can be derived through
\ 1\ 1\ 1(())
()
()
(())
(,)
(())
(( ))
jjjm
ii
A ij
j
m
ii
A im
ii
A ijPA paA e
PA e
PePA paA ePU ePA paA ePU e..
..
..
UUUUUjj
jj(10.7)
10.5 Use of BPN’s in Risk Assessment and Decision Analysis
Bayesian probabilistic networks can be used at any stage of a risk analysis, and may readily
substitute both fault trees and event trees in logical tree analysis. Furthermore, whereas
common cause or more general dependency phenomenon poses significant complications in
classical fault tree analysis this is not the case with Bayesian probabilistic nets. These nets are
basically designed to facilitate the modelling of such dependencies. Finally the Bayesian