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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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