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V
are said to be
d-connected
given
W
. This implies that the sets can exchange
information.
As the graphical part of a Bayesian network is meant to offer an explicit
representation of independence information in the associated joint probability
distribution
P
, we need to establish a relationship between
⊥
P
.Thisis
done as follows. We say that
G
is an
I-map
of
P
if any independence represented
in
G
by means of d-separation is satisfied by
P
, i.e.,
⊥
G
and
U
⊥
G
V
|
W
=
⇒
X
U
⊥
P
X
V
|
X
W
,
where
U
,
Z
and
W
are sets of nodes of the ADG
G
and
X
U
,
X
V
and
X
W
are
the sets of random variables corresponding to the sets of nodes
U
,
V
and
W
,
respectively. This definition implies that the graph
G
is allowed to represent
more dependence information than the probability distribution
P
, but never
more independence information.
A Bayesian network is a I-map model whose nodes represent random vari-
ables, and where missing arcs encode conditional independencies between the
variables. In short, it allows a compact representation of independence infor-
mation about the joint probability distribution
P
in terms of
local conditional
probability distributions
(
CPDs
), or, in the discrete case, in terms of
conditional
probability tables
(
CPTs
), associated to each random variable. This table de-
scribes the conditional distribution of the node given each possible combination
of values of its parents. The joint probability can be computed by simply multi-
plying the CPTs.
In a BN model the
local Markov property
holds, i.e., each variable is condi-
tionally independent of its non-descendants given its parents:
X
v
⊥
X
V
\de
(
v
)
|
X
pa
(
v
)
for all
v
∈
V
where
de
(
v
) is the set of descendants of
v
. The joint probability distribution
defined by a Bayesian network is given by
P
(
X
V
)=
v∈
V
P
(
x
v
|
X
pa
(
v
)
)
where
X
V
denote the set of random variables in the given model,
x
v
is a variable
in such set and
X
pa
(
v
)
represents the parent nodes of the variable
x
v
.BNisa
Bayesian network model with respect to G if its joint probability density function
can be written as a product of the individual density functions, conditional on
their parents.
Bayesian networks are commonly used in the representation of causal rela-
tionships. However, this does not have to be the case: an arc from node
u
to
node
z
does not require variable
x
z
to be causally dependent on
x
u
.Infact,
considering its graphical representation, it is the case that in a BN,
u
−→
z
−→
w
and
u
←−
z
←−
w
are equivalent, i.e., the same conditional independence constraints are enforced
by those graphs.
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