This recursive dependence structure

This recursive dependence structure can be represented by a directed graph
G
by
drawing an arrow from each vertex in
U
v
to
v
. As an immediate consequence of the
recursive factorization, the resulting graph is acyclic, that is, it does not contain any
loops.
On the other hand,
P
factorizes with respect to the undirected graph
G
m
which
is given by the class of complete subsets
D
=
{{
v
}∪
U
v
|
v
∈
V
}
. This graph can
be obtained from the directed graph
G
by completing all sets
{
v
}∪
U
v
and then
converting all directed edges into undirected ones. The graph
G
m
is called the moral
graph of
G
since it is obtained by “marrying all parents of a joint child”.
As an example, suppose that we want to describe the distribution of a genetic
phenotype (such as blood group) in a family. In general, we can assume that the
phenotype of the father (
X
) and that of the mother (
Y
) are independent, whereas
the phenotype of the child (
Z
) depend on the phenotypes of both parents. Thus the
joint density can be written as