Belief Function Theory - Information Fusion in Signal and Image Processing - page 111

phase, where a subset of D has to be chosen to maximize a certain criterion. From
here on, m , Bel and Pls refer to the mass, belief and plausibility functions obtained
after the combination.

In belief function theory, several decision rules are possible and are most of the
time applied to the choice of a singleton C i .

The maximum plausibility :

C i if Pls C i ( x )=max Pls C k ( x ) , 1 n ,

∈ ≤ ≤

x k [7.35]

this rule being optimal in the sense laid down by probabilistic criteria for mass func-
tions derived from probabilities [APP 91].

The maximum credibility :

C i if Bel C i ( x )=max Bel C k ( x ) , 1 n ,

x ∈ ≤ k ≤ [7.36]

which is equivalent to the maximum plausibility criterion in the case where the result
of the combination only involves singletons.

The maximum credibility without confidence interval overlap (without the risk of
an error) :

C i if Bel C i ( x ) max Pls C k ( x ) , 1 = i ,

x ∈ ≥ ≤ k ≤ n, k [7.37]

this last condition being particularly strict and possibly leading to no decision being
made.

The maximum credibility with discarding [MAS 97]:

C i if Bel C i ( x )=max Bel C k ( x ) , 1 n

x ∈ ≤ k ≤ [7.38]

and

Bel C i ,
which expresses the fact that the decision has to be unambiguous enough since the
condition will be met if the mass is very focused on C i .

Bel C i ( x )

≥

The maximum pignistic probability defined by [SME 90b]:

D, BetP C j =
C i ∈ A

m ( A )

1

)

∀ C j ∈ , [7.39]

| A | − m ( ∅

where refers to the number of elements in A thus making it possible to switch to

a probabilistic framework, which is often desired for making the decision (or the bet)
or for associating this decision with other probabilistic criteria, for example, in the
framework of Markov fields for spatial regularization criteria [TUP 99].

| A |