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this can be done on two levels in belief theory, either on credibilities, or on plausibili-
ties, thus leading to conclusions that are not necessarily equivalent.
Smets's axiom A6 does not imply that A and A are interchangeable, whereas this
property is explicitly used by Cox to obtain the second functional equation (equation
[A.6]), since subsets X can be involved in both ( m 1
m 2 )( A ).
Therefore, no complementarity relation regarding m can be obtained from it. This is
replaced by a duality relation between Bel and Pls.
m 2 )( A ) and ( m 1
Finally, axioms A7 and A8 are considered by Smets himself as technical axioms
used in the demonstrations. The regularity imposed on functions can be compared
with the regularity hypotheses formulated for Cox's two functional equations [A.5]
and [A.6].
These differences between the two theories have consequences on the three levels
that traditionally comprise the fusion process, i.e. the modeling of belief functions,
the combination of the functions determined from the information provided by several
sources and the final decision:
- first in the modeling phase, because this phase is strongly constrained by the two
functional relations (equations [A.5] and [A.6]) in probabilistic fusion, whereas belief
theory makes it possible to easily adapt to many situations (we mentioned the example
of sensors that only provide information regarding the union of two classes, without
distinguishing them);
- in the combination of belief functions, postulates impose Bayes' rule on the one
hand, Dempster's rule on the other hand, and their differences stem in particular from
the more flexible constraints imposed by Smets's conditioning rather than from Cox's
hypothetical conditioning;
- finally, in the decision making, i.e. the ultimate phase of the fusion process, dif-
ferences come mostly from comparing degrees of confidence, which give way to sev-
eral types of decision in the Dempster-Shafer theory.
B.4. Bibliography
[DUB 86] D UBOIS D., P RADE H., “On the Unicity of Dempster Rule of Combination”, Inter-
national Journal of Intelligent Systems , vol. 1, p. 133-142, 1986.
[GAC 93]
G ACÔGNE L., About a Foundation of Dempster's Rule,
Report, Laforia 93/27,
[KLA 92] K LAWONNN F., S CHWECKE E., “On the Axiomatic Justification of Dempster's
Rule of Combination”,
International Journal of Intelligent Systems , vol. 7, p. 469-478,
[SHA 76]
S HAFER G., A Mathematical Theory of Evidence , Princeton University Press, 1976.
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