Belief Function Theory - Information Fusion in Signal and Image Processing - page 107

note that the conflict between two identical mass functions is not equal to zero, and
that it increases as the mass becomes more spread out over the singletons.

At first, this combination rule was thought to be applicable only under the source
independent hypothesis. It has been shown [QUI 89, QUI 91] that the rule can still be
applied without this hypothesis, by relying on the analogy with random closed sets.
In less technical and more philosophical terms, independence in the framework of
belief functions should not be understood in a statistical sense, but instead in a more
“cognitive” sense [SME 93]. This is referred to as cognitive independence. Imagine,
for example, that we wish to combine the opinions of experts. They are likely not to
be statistically independent (if they are experts in the same field), but we can expect
them to be cognitively independent, i.e. each one makes up his own opinion with-
out consulting the others. This is the type of independence Dempster's rule applies
to, which results in the non-idempotence of the rule, causing the reinforcement of
identical mass functions. Under the dependence hypothesis, we would want an idem-
potent rule instead. We will continue with these considerations with fuzzy set theory
in Chapter 8.

When the functions m , Bel and Pls are probabilities (i.e. when only the focal
elements are singletons), Dempster's combination law is consistent with the traditional
probability laws. In this light, probabilities are shown as the limit of belief theory,
when there is no ambiguity or imprecision and only the uncertainty of the data needs
to be taken into account.

Dempster's rule has a conjunctive behavior, since it provides focal elements that
are the intersections of the focal elements of the initial mass functions. Therefore, it
reinforces focusing and decreases the length of the confidence intervals [Bel , Pls].

In practice, the combination calculation is conducted by laying down the intersec-
tion table of the focal elements. For example, if m 1 C 2 (typical of a

source no longer capable to tell two classes apart) and C 3 , and m 2 to C 1 and C 2 ∪

pertains to C 1 ∪

C 3 ,

the focal elements of m 1 ⊕ m 2 are given by the following intersection table:

C 1 ∪ C 2 C 3

C 1 C 1 ∅

C 2 ∪ C 3 C 2 C 3

In this case, the focal elements are simply the singletons and the empty set. This
example illustrates how conjunctive combination reduces the imprecision and solves
(or generally reduces) the ambiguity of each source.