52
D. Krpelík and T. Basu
sup
Z∈Z,Z≤Y
E(Z) =
inf
p∈(R
+
0 )
n
n
i p i =1
∀X j ∈K:
n
i=1 p i X j (ω)=E(X j )
n
i
p i Y (ω i ),
which effectively means that we are extremising the expectation over some set
of admissible distributions p, which would yield the known expectations for all
X n ∈ K.
2.4 Imprecise Probabilities
Compared with the point-valued and imprecise specification of the parameter in
Sects. 2.2.1 and 2.2.2, we take the approach for generalising the precise probability
theory by considering sets of precise distribution models for a RV. Analyses in
the IP framework are then conducted by analysing the properties of these sets.
But specifying such sets and working with them are not always as simple as in
Sect. 2.2.4.
In the rest of the chapter, we are going to introduce the underlying mathematical
structures needed for the treatment of imprecisely specified RVs.
2.4.1 A Set of Measures
We begin by introducing an extension to measure-theoretic probability given by
Weichselberger [43] and similarly by Walley [41]. The idea is to assign to every
event E ∈ Ω a pair of real numbers ∈ [0, 1], which represent the lower and
upper bounds for the probability of that event. These bounds will reoccur within
the rest of the theory of imprecise probabilities. They also have an epistemological
interpretation—the lower bound measures the evidence supporting the occurrence
of E, whereas the upper bound measures the evidence, which contradicts E. Their
difference is the measure of our ignorance.
Definition 2.6 An interval-valued set function P on a measurable space
(Ω, A) will be called a R-probability if:
• ∀E ∈ A : P (E) = [L(E), U (E)] s.t. 0 ≤ L(E) ≤ U(E) ≤ 1.
• The set M := {p : p is K-probability , ∀E ∈ A : p(E) ∈ P (E)} = ∅.
(continued)
D. Krpelík and T. Basu
sup
Z∈Z,Z≤Y
E(Z) =
inf
p∈(R
+
0 )
n
n
i p i =1
∀X j ∈K:
n
i=1 p i X j (ω)=E(X j )
n
i
p i Y (ω i ),
which effectively means that we are extremising the expectation over some set
of admissible distributions p, which would yield the known expectations for all
X n ∈ K.
2.4 Imprecise Probabilities
Compared with the point-valued and imprecise specification of the parameter in
Sects. 2.2.1 and 2.2.2, we take the approach for generalising the precise probability
theory by considering sets of precise distribution models for a RV. Analyses in
the IP framework are then conducted by analysing the properties of these sets.
But specifying such sets and working with them are not always as simple as in
Sect. 2.2.4.
In the rest of the chapter, we are going to introduce the underlying mathematical
structures needed for the treatment of imprecisely specified RVs.
2.4.1 A Set of Measures
We begin by introducing an extension to measure-theoretic probability given by
Weichselberger [43] and similarly by Walley [41]. The idea is to assign to every
event E ∈ Ω a pair of real numbers ∈ [0, 1], which represent the lower and
upper bounds for the probability of that event. These bounds will reoccur within
the rest of the theory of imprecise probabilities. They also have an epistemological
interpretation—the lower bound measures the evidence supporting the occurrence
of E, whereas the upper bound measures the evidence, which contradicts E. Their
difference is the measure of our ignorance.
Definition 2.6 An interval-valued set function P on a measurable space
(Ω, A) will be called a R-probability if:
• ∀E ∈ A : P (E) = [L(E), U (E)] s.t. 0 ≤ L(E) ≤ U(E) ≤ 1.
• The set M := {p : p is K-probability , ∀E ∈ A : p(E) ∈ P (E)} = ∅.
(continued)