Titel_SS06

Bridge1

G B

Bridge 2 G 0.81+0.0225+0.0225+0=0.855 0.0225+0.0225=0.045

B 0+0.0225+0.0225+0=0.045 0+0.0225+0.0225+0.01=0.055

Table 10.7: Discrete joint probability distribution of the states of Bridge 1 and Bridge 2.

Based on the joint probability distribution of the states of the two bridges given in Table 10.7
the probability table given in Table 10.8 has been derived.

BR2 BR1=B

G 0.045/(0.045+0.055)=0.45

B 0.055/(0.045+0.055)=0.55

Table 10.8: Discrete conditional probability distribution for the different states of Bridge 2 given
evidence concerning the state of Bridge 1.

It is easily seen that without any evidence introduced the probability of either bridge being in
the state B is only 0.1. The introduction of evidence about the condition of one bridge thus has
a significant effect on the probability of the states of the other bridge. As earlier discussed this
effect was already intuitively expected but the present example has shown that the Bayesian
Networks are able to capture this behaviour. Furthermore, the example illustrates how the
Bayesian Networks facilitate the analysis of the causal relationships in terms of the joint
probability distribution of the states of the variables represented by the network. It should,
however, be noted that the considered example is simple if not even trivial for two reasons,
first of all because the number of variables and the number of different variable states is small
and secondly because the number of edges in the considered network is small. Generally
speaking the numerical effort required for the analysis of Bayesian Networks grows
exponentially with the number of variables, variable states and edges. For this reason
techniques to reduce this effort are required.

The reader is referred to e.g. Jensen (1996) for details on how to keep the numerical treatment
of Bayesian Nets tractable; however, a few simple approaches for this will be illustrated in the
following.

The first approach is to wait with the application of the chain rule on the Bayesian Net until
evidence has been introduced. Introducing evidence corresponds to fixing (or instantiating)
the states of one or more variables in the Bayesian Net. For each instantiated variable the
dimension of the probability table of the Bayesian Net is reduced by a factor corresponding to
the number of states of the instantiated variable. The second approach which is mentioned
here is called bucketing. The idea behind this approach is to benefit from the fact that the
probability distribution for one or several variables in a Bayesian Net is achieved by
marginalization of the probability distribution function of the universe represented by the
Bayesian Net. By rearranging the product terms in the chain rule for the Bayesian Net such