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Fig. 2.1
Graphical separation, conditional independence, and probability decomposition for the
three
fundamental connections
(from
top
to
bottom
):
converging connection
,
serial connection
,
and
diverging connection
2.1.2 Fundamental Connections
Consider the
fundamental connections
(
Jensen
,
2001
) shown in Fig.
2.1
,thethree
possible configurations of three nodes and two arcs. In the
convergent connection
or
v-structure
, node
C
has incoming arcs from
A
and
B
, thus violating both conditions
in Definition
2.2
. Therefore, we conclude that
C
does not d-separate
A
and
B
.This
in turn implies that
A
and
B
are not independent given
C
, and since
Π
A
=
{
∅
}
,
Π
B
=
{
∅
}
,and
Π
C
=
{
A
,
B
}
,wehave
P
(
A
,
B
,
C
)=
P
(
C
|
A
,
B
)
P
(
A
)
P
(
B
)
(2.6)
from the Markov property introduced in Eq.
2.4
. From the above expression, it is
evident that
C
depends on the joint distributions of
A
and
B
. Therefore,
A
and
B
are
not conditionally independent given
C
. On the other hand,
A
and
B
are independent
given
C
in the
serial
and
diverging
connections since the conditions in Definition
2.2
are satisfied in these cases. For the serial connection, we have
Π
A
=
{
∅
}
,
Π
B
=
{
C
}
,
and
Π
C
=
{
A
}
; therefore,
(
,
,
)=
(
|
)
(
|
)
(
)
.
P
A
B
C
P
B
C
P
C
A
P
A
(2.7)
For the diverging connection, we have
Π
=
{
C
}
,
Π
=
{
C
}
,and
Π
=
{
∅
}
;
A
B
C
therefore,
P
(
A
,
B
,
C
)=
P
(
A
|
C
)
P
(
B
|
C
)
P
(
C
)
.
(2.8)
2.1.3 Equivalent Structures
From Fig.
2.1
, it should also be noted that the
serial
and
diverging
connections
result in equivalent factorizations; each can be obtained from the other with re-
peated applications of Bayes' theorem. Such probabilistically equivalent structures
are known as
Markov equivalent
structures. Since equivalence is symmetric, re-
flexive, and transitive, each set of equivalent structures forms an
equivalence class
.
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