56
D. Krpelík and T. Basu
Weichselberger’s treatment provides a useful framework to define the lower
and upper probabilities and even the desired extending properties for partial
specifications. Probability bounds given for all the elementary events E ∈ A ⊂ Ω
can be extended into bounds for arbitrary event E ∈ Ω by solving a linear
optimisation problem.
For the derived RVs Y = f (X), we may calculate the imprecise probabilities
that they will obtain a value in an element of their respective σ -algebras, similarly
as in the precise case through Eq. (2.7) and Theorem 2.1. The imprecision in the
distributions will also manifest in the imprecision in the expected values. We can
calculate the lower and upper expectations, but for the structures introduced in this
section, we can only do so through by optimising the expected value over the credal
set.
2.4.2 Capacities
Capacities originated in the work of Choquet [10] on the generalisation of measure
theory for non-additive measures. They further provide useful properties and
structure to IP models, which allows us to simplify the specification of the bounds
for arbitrary event E ∈ A and the expectations of RVs.
Definition 2.10 Let (Ω, A) be a measurable space. A set function g : A →
R is called a capacity if it is monotone, i.e.
∀A, B ∈ A : A ⊂ B ⇒ g(A) ≤ g(B).
A capacity g is further called super-additive if
∀A, B ∈ A : A ∩ B = ∅ ⇒ g(A ∪ B) ≥ g(A) + g(B).
If the inequality is reversed, it is instead called sub-additive.
Note that if the structure M of an F-probability is closed (i.e. arginf belongs to
M), then both L and U are super- and sub-additive capacities, respectively.
Definition 2.11 A capacity g is said to be n-monotone if for any collection
E n ⊂ A of n elements
(continued)
D. Krpelík and T. Basu
Weichselberger’s treatment provides a useful framework to define the lower
and upper probabilities and even the desired extending properties for partial
specifications. Probability bounds given for all the elementary events E ∈ A ⊂ Ω
can be extended into bounds for arbitrary event E ∈ Ω by solving a linear
optimisation problem.
For the derived RVs Y = f (X), we may calculate the imprecise probabilities
that they will obtain a value in an element of their respective σ -algebras, similarly
as in the precise case through Eq. (2.7) and Theorem 2.1. The imprecision in the
distributions will also manifest in the imprecision in the expected values. We can
calculate the lower and upper expectations, but for the structures introduced in this
section, we can only do so through by optimising the expected value over the credal
set.
2.4.2 Capacities
Capacities originated in the work of Choquet [10] on the generalisation of measure
theory for non-additive measures. They further provide useful properties and
structure to IP models, which allows us to simplify the specification of the bounds
for arbitrary event E ∈ A and the expectations of RVs.
Definition 2.10 Let (Ω, A) be a measurable space. A set function g : A →
R is called a capacity if it is monotone, i.e.
∀A, B ∈ A : A ⊂ B ⇒ g(A) ≤ g(B).
A capacity g is further called super-additive if
∀A, B ∈ A : A ∩ B = ∅ ⇒ g(A ∪ B) ≥ g(A) + g(B).
If the inequality is reversed, it is instead called sub-additive.
Note that if the structure M of an F-probability is closed (i.e. arginf belongs to
M), then both L and U are super- and sub-additive capacities, respectively.
Definition 2.11 A capacity g is said to be n-monotone if for any collection
E n ⊂ A of n elements
(continued)