Probabilities: A Historical Perspective - Information Fusion in Signal and Image Processing - page 272

We should also mention the works of Gacôgne, which have led to a justification,
in the specific cases where the frame of discernment is reduced to two elements, based
on the concept of accentuation [GAC 93] 1 .

The works of Smets offer the most general justification, as far as we know, and
his arguments will be described here. Furthermore, his method is similar to that of
Cox (see Appendix A). Work similar to that of Smets has been done by Klawonn and
Schwecke [KLA 92].

B.1. Smets's axioms

The first observation Smets made involves the indifference principle (or principle
of insufficient reason). Assigning the same probability to every simple event implies
that different probabilities will be assigned to union of events, which, to Smets, does
not correspond to indifference. This should instead be expressed by the existence of a
constant that is positive or equal to zero and such that:

= D, Bel( A )= c, [B.1]
where D refers to the frame of discernment. This is obviously impossible with proba-
bilities, but it is not in the Dempster-Shafer context with credibilities, since we have:

∀ A ⊂ D, A

A ∩ B = ∅ = ⇒ Bel( A ∪ B ) ≥ Bel( A )+Bel( B ) [B.2]

hence c 2 c and therefore c =0. The mass function representing indifference (or

complete lack of knowledge) is therefore defined by:

≥

m ( D )=1and ∀ A = D, m ( A )=0 , [B.3]

1. Gacôgne shows that the Dempster-Shafer rule (equation [7.26]), in the case where the frame
of discernment is reduced to a proposition P and its opposite P , can be inferred from the
concept of accentuation. An accentuation function is such that it reduces the degrees of confi-
dence smaller than 0.5 and increases those greater than 0.5, thus making them more like binary
degrees (this is equivalent to the concept of reinforcement found in the algebraic theory of
ordered semigroups as well as in fuzzy set theory). The rational function of [0 , 1] in [0 , 1]
with the lowest degree that is an accentuation function is defined by x 2 / (2 x 2 − 2 x +1), and
that is the one that can be used to show the analogy with Dempster-Shafer. This concept is
then generalized to pairs ( x, y ), characterizing a proposition P , such that 0 ≤ x ≤ y ≤ 1
(referred to as “obligation” and “eventuality”, hence similar to the concept of credibility and
plausibility or of necessity and possibility). The accentuation of such a couple is defined by:
#( x, y ) = ((2 xy − x 2 ) / (1 − 2 x +2 xy ) ,y 2 / (1 − 2 x +2 xy )). These two values corre-
spond exactly to those we would obtain by combining, using the Dempster-Shafer rule (equa-
tion [7.26]), the credibility (respectively, the plausibility) of a proposition P with itself. If we
now have at our disposal two games with the measures ( x, y ) and ( x ,y ) on P , the same type
of reasoning leads to the justification of the Dempster-Shafer rule [GAC 93]. We would then be
left with having to generalize this approach to more complex spaces of discernment.