Image Processing Reference
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a response to traditional probability theories, where, in order to achieve objectivity,
certain types of information have to be dismissed.
Postulates 2 and 3 lead to the use of real numbers for representing and comparing
degrees of confidence: a single real number is necessary and sufficient for representing
a degree of confidence and the switch from true to false is continuous.
Postulate 1 leads to the existence of functional relations between the degrees of
confidence.
Postulate 4 imposes that traditional, deductive symbolic logic is a specific case.
Postulate 5 leads to hypothetical conditioning: the degree of confidence in a propo-
sition
A
is only know conditionally to a state of knowledge
e
that represents informa-
tion related to the confidence in
A
and assumed (or believed) to be true. Such a degree
of confidence is denoted by [
A
|
e
].
The consistency postulate and hypothetical conditioning impose the existence of a
functional equation
T
relating [
AB
e
] (degree of confidence in “
A
and
B
”forthe
state of knowledge
e
) and at least two of the quantities [
A
|
|
e
], [
A
|
Be
], [
B
|
e
],
[
B
Ae
], and the existence of a functional relation
S
between the degrees of confi-
dence in a proposition [
A
|
e
] and in its negation [
A
|
|
e
].
Paris [PAR 95], in a rigorous demonstration of the works of Cox, insists on a
hypothesis that is often omitted but is crucial to the demonstration:
[0
,
1]
3
,
∀
(
α, β, γ
)
∈
∀
ε>
0
,
∃
A, B, C, D,
verifying the consistency postulate
/
[
D
α
<ε,
[
C
β
<ε,
[
B
γ
<ε.
|
ABC
]
−
|
AB
]
−
|
A
]
−
In particular, this hypothesis cannot be verified in a finite frame of reference.
A.3.2.
First functional equation
For the relation
T
, 11 functions are possible (6 with 2 arguments, 4 with 3 argu-
ments and 1 with 4 arguments). Because the roles of
A
and
B
are symmetric, these
functions can be reduced to 7. With the rest of the arguments, certain types of functions
can be eliminated by examining specific cases that lead to absurdities.
If the state of knowledge
e
stipulates that “
A
and
B
are independent”, then
[
A
|
e
]=[
A
|
Be
]. Among the 7 forms of
T
, the one that is a function of [
A
|
e
] and
[
A
|
Be
] is then a function of [
A
|
e
] only and no longer depends on
B
. This form
must therefore be eliminated.
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