Belief Function Theory - Information Fusion in Signal and Image Processing - page 104

7.4. Conjunctive combination

7.4.1. Dempster's rule

l ) be the mass function defined for the source j . The conjunctive
combination of the mass functions is conducted according to Dempster's orthogonal
rule [SHA 76, SME 90a], defined

Let m j ( j =1 ···

∀ A ⊆ D by:

m l ( A )=
B 1 ∩···∩ B l = A

m 1 ⊕

m 1 B 1 m 2 B 2 ··· m l B l .

m 2 ⊕···⊕ [7.26]

Axiomatic justifications of this rule can be found in [SME 90a]. The differences
between these axioms and those of Cox [COX 46] (which are used to justify the prob-
ability rules) explain the origins of the differences between the two theories [BLO 95].
These aspects are discussed in Appendix B.

7.4.2. Conflict and normalization

In the non-normalized equation [7.26], the mass assigned by combination to the
empty set is usually not equal to zero. It is often interpreted as the conflict between the
sources. Let us note that this conflict measurement is not absolute, but instead depends
on the modeling (particularly of the distribution of masses among the different subsets
of D ). There are two essential sources of conflict: either the sources are not reliable, or
they provide information about different phenomena. In the first case, it is acceptable
to combine the sources and a solution for taking the conflict into account is to weaken
the sources according to their reliability. We will discuss this further later on. In the
second case, combination makes no sense. Methods for regrouping sources according
to the phenomena they observe have been suggested, with the objective of combining
only the sources within each group. These groups are calculated in such a way as to
minimize the conflict within each group [MIL 01, SCH 93].

In an open world hypothesis, a non-zero mass on the empty set can also represent
a solution that was not predicted in D . Under a closed world hypothesis, where every-
thing that is possible is represented in D , this interpretation is not acceptable, which
leads us to normalizing the result of the combination in the following form 2 :

m l ( A )= B 1 ∩···∩ B l = A m 1 B 1 ···

m l B l

m 1 ⊕···⊕

− B 1 ∩···∩ B l = ∅ m 1 B 1 ··· m l B l

, [7.27]

1

2. This normalized form is Dempster's rule in its strict sense [SHA 76]; the non-normalized rule
was suggested later [SME 90a] but seems preferable today for most applications.