2 Introduction to Imprecise Probabilities
57
Definition 2.11 (continued)
g
⎛
⎝
E∈E n
E
⎞
⎠ ≥
E⊂E n
(−1)
|E|+1 g
E∈E
E
.
If g is monotone for every n ∈ N, then it is called ∞-monotone.
Corollary
Any n-monotone capacity is also n > m-monotone.
Two-monotone capacities are coherent. A pair of super- and sub-additive capacities, such that the sub-additive one dominates the super-additive one, constitute an
F-probability.
Definition 2.12 For a super-additive capacity g defined on a finite space Ω,
we define, for every event E ⊂ Ω, a function m g : 2 Ω → R, the möbius
inverse, as
m g (E) :=
A⊂E
(−1)
|E\A| g(A).
The benefit is that an inverse mapping exists, which enables us to reconstruct the
capacity from its möbius inverse as
g(E) =
A⊂E
m g (A).
(2.12)
The dual capacity, the upper probability, can be reconstructed as
g
∗ (E) =
{A∈2 Ω :A∩E =∅}
m g (A).
(2.13)
A special class of models is composed of ∞-monotone lower probabilities on
finite spaces. Their möbius inverses (aka the mass functions) are non-negative for
every event. Conversely, any normalised (
m = 1) non-negative function m :
2 Ω → R with a finite support induces ∞-monotone lower and upper probabilities
by Eqs. (2.12) and (2.13), respectively. This is explicitly exploited in the evidence
theory (see Chap. 17).