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The region of the universe
U
that is characterized by an increased probabilistic
connection with the target set
X
ↆ
(
)
U
, relative to the prior probability
P
X
, is called
the absolute positive region
of the set
X
, denoted as
POS
∗
(
X
)
:
POS
∗
(
X
)
=∪{
E
:
P
(
X
|
E
)>
P
(
X
)
}
.
(6.9)
In the absolute positive region of
X
, the likelihood of an object belonging to the
target set is higher than in the whole universe
U
, but in practice that increase may
be not sufficient from an application perspective.
Similarly, the
absolute negative region
,
NEG
∗
(
, of the target set
X
is an area
of the universe
U
characterized by reduced likelihood of an object being a member
of the target set
X
:
X
)
NEG
∗
(
X
)
=∪{
E
:
P
(
X
|
E
)<
P
(
X
)
}
.
(6.10)
The above definitions provide the basis of the Bayesian rough set model [
13
,
30
].
6.3 Dependencies in Approximation Spaces
The probabilistic connections between elementary sets and the target set, and between
definable sets and the target set in the approximation spaces can be quantified by using
different dependencymeasures [
24
,
33
]. Some of these measures are reviewed below.
6.3.1 Absolute Certainty Gain
Absolute certainty gain
, denoted as
gabs
, evaluates the degree of one-directional
dependency between any two sets. In the simplest case, it is a single-directional
dependency measure representing the degree of change of the probability of mem-
bership in the set
X
for an object belonging to the elementary set
E
. The absolute
certainty gain is defined by:
gabs
(
X
|
E
)
=|
P
(
X
|
E
)
−
P
(
X
)
|
,
(6.11)
where
is the absolute value function.
The above definition can be extended to any definable set
Y
. The absolute certainty
gain between the subsets
X
and
Y
can be computed directly from the available
probabilistic knowledge based on the formula below, where the summation is over
all elementary sets forming the definable set
Y
:
|
.
|
)
=
|
ʣ
E
ↆ
Y
P
(
E
)
P
(
X
|
E
)
−
P
(
X
)ʣ
E
ↆ
Y
P
(
E
)
|
gabs
(
X
|
Y
.
(6.12)
ʣ
E
ↆ
Y
P
(
E
)
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