Belief Function Theory - Information Fusion in Signal and Image Processing - page 97

Given a mass function m ,theBel function defined by:

Bel( A )=
B ⊆ A,B = ∅

2 D , ∀ A ∈ m ( B ) [7.3]

is a belief function. Conversely, from a belief function defined as a totally increasing
function (inequality [7.2]) such that Bel(

∅

)=0, Bel( D )=1, we can define a mass

function by:

m ( A )=
B ⊆ A

1) | A − B | Bel( B ) .

2 D ,

∀ A ∈ ( − [7.4]

This mass function then verifies equation [7.3].

The belief function measures the total confidence placed in the set A . The empty
set is excluded from the sum because it would otherwise be found in both the evalua-
tion of A and the evaluation of A C A C ).

( ∅⊂ A and ∅⊂

Thus, having a zero mass on a subset A does not mean that this set is impossible,
simply that we are not capable of assigning a level precisely to A , since we could have
non-zero masses on subsets of A , which would lead us to Bel( A ) =0. This comment

is very important for modeling because it allows us not to assign confidence values
when we are not able to do so (this way, we are not forcing information where none is
available).

In the open world hypothesis, we have:

Bel( D )=1 − m ( ∅ ) . [7.5]

A plausibility function Pls is also a function of 2 D into [0 , 1] defined by:

Pls( A )=
B ∩ A = ∅

Bel A C . 2 D ,

∀ A ∈ m ( B )=1 − [7.6]

More generally, in order to account for the possibility of dealing with an open
world, we have:

Pls( A )=
B ∩ A = ∅

Bel( A C ) . m ( B )=Bel( D ) − [7.7]

Plausibility measures the maximum confidence that can be placed in A . This func-
tion has a natural interpretation in the transferable belief model [SME 90a] in which
additional information is considered to allow for the transfer of belief to more precise