Image Processing Reference
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subsets. Plausibility then represents the maximum belief that could potentially be
assigned to a subset
A
if we learned, for example, that the solution is in
A
(all of
the confidence placed in a subset
B
intersecting
A
is then transferred to
A
in order to
lower to 0 the confidence in
A
C
).
We have the following properties:
2
D
,
∀
A
∈
Pls(
A
)
≥
Bel(
A
)
,
[7.8]
2
D
,
Bel(
A
)+Bel(
A
C
)
∀
A
∈
≤
1
,
[7.9]
2
D
,
Pls(
A
)+Pls(
A
C
)
∀
A
∈
≥
1
,
[7.10]
2
D
,
Bel(
A
)+Bel(
A
C
)=1
∀
A
∈
⇐⇒
Bel(
A
)=Pls(
A
)
.
[7.11]
The interval [Bel(
A
)
,
Pls(
A
)] is referred to as the confidence interval and its length
measures the absence of knowledge we have of an event
A
and its complement.
>
1), then
the three functions
m
, Bel and Pls are equal and are a probability. They are referred
to as Bayesian mass functions. In more complex situations, this is not the case and
there is no direct equivalence with probabilities. Functions similar to credibility and
plausibility functions could be obtained, for example, from probabilities conditional
to pessimistic and optimistic behaviors respectively, but their formalization would be
much more difficult than what belief function theory has to offer.
If we assign masses only to the simple hypotheses (
m
(
A
)=0for
|
A
|
Among the distinctive mass functions, there is a category for simple support func-
tions, for which all of the mass is assigned to a non-empty subset
A
and to a set of
discernment
D
:
m
(
A
)=
s
m
(
D
)=1
−
s
m
(
B
)=0 for any
B, B
=
A, B
=
D,
with
s
∈
[0
,
1].
If
s
is equal to 0, then the entire mass is assigned to
D
. This function represents
the total lack of knowledge, in the sense that no subsets can be distinguished.
The possibility of assigning masses to the composite hypotheses and therefore to
work on 2
D
rather than on
D
constitutes one of the advantages of this theory because
it allows for very flexible and rich modeling, particularly of ambiguity or hesitation
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