called a directed graph. The relations between the variables being expressed in terms of
family relations, such that when the link goes from variable A to variable B then the variable A
is a parent to B and B is a child of A. The variables can in principle have any number of
discrete states or a continuous sample space but can, however, only attain one realisation at
one time.
Networks are categorised in accordance with their configuration. In Figure 10.2 a serial
connection is illustrated. A has an influence on B which again has an influence on C. If
evidence is introduced about the state of A this will influence the certainty about the state of B
which then influences the certainty about the state of C. However, if the state of B is known
with certainty the channel is blocked and the variables A and C become conditional
independent. A and C are d-separated given B. Therefore evidence can be transmitted through
a serial connection only if the states of the variables in the connection are unknown.
A B CFigure 10.2: Illustration of a serially connected network.
In Figure 10.3 a diverging connection is illustrated. The information about any of the children
of A can influence the other children as long as the state of the parent A is not known with
certainty. The children B, C and D are d-separated given A.
BC DAFigure 10.3: Illustration of a diverging network.
A converging connection is illustrated in Figure 10.4. This type of connection requires a little
more care. As long as no evidence is available regarding the state of the child A except what
may be inferred from its parents B, C and D the parents remain independent. No information
is transferred through the child variable A. However, as soon as evidence is introduced, i.e.
evidence about the state of the variable A or any one of the parents B, C and D, all the parents
become dependent.
This phenomenon is called conditional dependence.
If the information about the state of a variable is certain it is usually referred to as hard
evidence, otherwise as soft. Sometimes hard evidence is also referred to as instantiation.
Blocking in the case of serial or diverging connections requires hard evidence, whereas
opening in the case of converging connections requires either soft or hard evidence. If
variables are not d-separated they are denoted d-connected.