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where
p
(
x
i
|
Pa
i
)(
i
=1, 2, …,
n
) is the local probabilistic distribution in formula
(6.32). The pair (S,
). If
Bayesian network is constructed merely based on prior information, the
probabilistic distribution is Bayesian, or subjective. If Bayesian network is
constructed purely based on data, the distribution is physical, or objective.
To construct Bayesian network, we should do the following work:
Step 1
Determine all the related variables and their explanations. To do so, we
need: (1) Determine the objective of the model, or make a reasonable explanation
of given problem; (2) Find as many as possible problem related observations, and
determine a subset that is worth of constructing model; (3) Translate these
observations into mutual exclusive and exhaustive state variables. The result of
these operations is not unique.
Step 2
Construct a directed acyclic graph, which expresses conditional
independent assertion. According to multiplication formula, we have:
P
) represents the joint probabilistic distribution
p
(
X
∏
n
p x
(
|
x x
,
,
?
,
x
−
)
p
(
X
)=
1
2
i
1
i
i
=
1
p x
(
)
p x
(
|
x
)
p x
(
|
x
,
x
)
?
p x
(
|
x
,
x
,
?
,
x
)
=
(6.33)
1
2
1
3
1
2
n
1
2
n
−
1
For any variable
X
, if there is a subset
i
⊆{
x
1
,
x
2
,
?
,
x
i
-1
}, so that
x
i
and {
x
1
,
2
,
?
,
x
i
-1
}\
i
are conditional independent. That is, for any given
X
, the
following equation holds.
x
p(x
i
|x
1
,x
2
,
?
, x
i-1
)=p(x
i
|
ʩ
i
),
(
i=1,2,
?
,n
) (6.34)
According to formula (6.33) and (6.34) we have
n
∏
p x
(
|
π
)
p
(
x
)=
. The
i
i
i
=
1
variable set (
1
, …,
n
) corresponds to the parent set (
Pa
1
,
?
,
Pa
n
). So the
above equation can also be written as
n
∏
p x
(
|
Pa
)
. To determine the
structure of Bayesian network, we need to (1) sort variables
p
(
X
)=
i
i
i
=
1
x
1
,
x
2
,
?
,
x
i
;
(2)determine variable set (
1
,
?
,
n
) that satisfies formula (6.34).
Theoretically, finding a proper conditional independent sequence from n
variables is a combination explosion problem, for it will require comparison
among n! different sequences. In practice, casual relation is often used to solve
this problem. Generally, casual relation will correspond to conditional
independent assertion. So we can find a proper sequence by adding arrowed arcs
from reason variables to result variables.
Step 3
Assign local probabilistic distribution
x
i
|
Pa
i
). In the discrete case, we
need to assign a distribution for each variable on each state of its parent nodes.
Obviously, the steps above may be intermingled but not purely performed in
sequence.
p
(
