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the random variables in
X
and the nodes
V
of
G
, such that for all disjoint subsets
A
,
B
,
C
of
A
⊥
P
B
|
C
⇐
=
A
⊥
G
B
|
C
.
(2.1)
Similarly,
G
is a
dependency map
(D-map) of
P
if
X
we have
⊥
|
=
⇒
⊥
G
B
|
.
A
P
B
C
A
C
(2.2)
G
is said to be a
perfect map
of
P
if it is both a D-map and an I-map,
A
⊥
P
B
|
C
⇐⇒
A
⊥
G
B
|
C
,
(2.3)
and in this case,
P
is said to be
isomorphic
or
faithful
to
G
.
The correspondence between the structure of the DAG
G
and the conditional
independence relationships it represents is elucidated by the
directed separation
criterion (
Pearl
,
1988
), or
d-separation
, as discussed below.
Definition 2.2 (D-separation).
If
A
,
B
,and
C
are three disjoint subsets of nodes in
a DAG
G
,then
C
is said to
d-separate
A
from
B
, denoted
A
C
, if along every
sequence of arcs
1
between a node in
A
and a node in
B
there is a node
v
satisfying
one of the following two conditions:
⊥
G
B
|
1.
v
has converging arcs (i.e., there are two arcs pointing to
v
from the adjacent
nodes in the path) and none of
v
or its descendants (i.e., the nodes that can be
reached from
v
)arein
C
.
2.
v
is in
C
and does not have converging arcs.
The
Markov property
of Bayesian networks, which follows directly from d-
separation, enables the representation of the joint probability distribution of the
random variables in
X
(the
global distribution
) as a product of conditional prob-
ability distributions (the
local distributions
associated with each variable
X
i
). This
is a direct application of the
chain rule
(
Korb and Nicholson
,
2010
). In the case of
discrete random variables, the factorization of the joint probability distribution P
X
is given by
p
i
=
1
P
X
i
(
X
i
|
Π
X
i
)
,
P
X
(
X
)=
(2.4)
where
Π
X
i
is the set of the parents of
X
i
; in the case of continuous random variables,
the factorization of the joint density function
f
X
is given by
p
i
=
1
f
X
i
(
X
i
|
Π
X
i
)
.
f
X
(
X
)=
(2.5)
Similar results hold for mixed probability distributions (i.e., probability distributions
including both discrete and continuous random variables).
1
They are often referred to as
paths
, using the more general definition that disregards arc
directions.
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