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Axiom
A1
, which expresses the dependence between degrees of confidence and
their combinations, is not as strict as Cox's postulate. Indeed, the consistency postulate
implies the existence of a relation defining the degree of confidence in
AB
which
involves only the propositions
A
and
B
, in the form of degrees of confidence assigned
to [
A
A
] and [
A
]) but not to other propositions. Smets's axiom,
which is more general, corresponds to the possibility provided by the Dempster-Shafer
theory to deal with subsets and not simply with singletons.
|
B
] and [
B
] (or [
B
|
Axioms
A2
,
A3
and
A5
correspond to properties of classical propositional logic.
Cox's postulates (particularly postulate 4) also imply that deductive logic exists as a
specific case. Therefore, the two methods coincide with each other on this point. These
axioms are used in Cox's method to eliminate certain forms of functional relations
between [
AB
e
] and the other degrees of confidence, in order to keep only the form
that is consistent with deductive logic:
|
e
]=
T
[
A
e
]
=
T
[
B
e
]
.
[
AB
|
|
Be
]
,
[
B
|
|
Ae
]
,
[
A
|
[B.11]
Likewise, these axioms are used in Smets's demonstration to eliminate depen-
dences and prove that (
q
1
⊕
q
2
)(
A
) at first only depends on
A
, and on
q
1
(
X
) and
q
2
(
X
) for
X
⊆
A
; then, in a second phase, only on
q
1
(
A
) and
q
2
(
A
).
Axiom
A4
(conditioning) expresses an idea that is very similar to the hypothetical
conditioning obtained from Cox's fifth postulate. The main difference is that condi-
tioning, this time, is expressed more as a compatibility relation than as a conditional
probability.
There is no equivalent to Cox's postulate 3 (universality) in Smets's axioms. This
is justified by the very basis of belief theory, in which propositions are character-
ized by two numbers (credibility and plausibility) instead of just one, and in which
well-defined propositions are allowed not to have a degree of confidence assigned to
them
3
. This flexibility is helpful for solving problems related to lack of information:
if a source is not capable of providing information about
A
, but provides some, for
example, about
A
B
, this situation is naturally taken into account by belief function
theory by assigning a mass to
A
∪
B
and not to
A
, whereas it would often require
including hypotheses or models in probability theory in order to be able to assign a
degree of confidence to
A
. From the perspective of comparing degrees of confidence,
∪
3. This can be done, for example, by assigning a zero mass to this proposition
A
. This does not
mean, however, that a zero confidence is attributed to
A
, since the credibility Bel(
A
) and the
plausibility Pls(
A
) are not necessarily equal to zero because non-zero masses can be assigned
to propositions
B
A ∩ B
=
∅
such that
. This simply means that no degree of confidence is
assigned specifically to
A
.
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