2 Introduction to Imprecise Probabilities
53
Definition 2.6 (continued)
We will call the tuple R := (Ω, A, L, U) a R-probability field. The set M
from the second axiom is called the structure (Weichselberger) or the credal
set (Walley) of R.
The letter R- indicates reasonable. It corresponds to the property of avoiding sure
loss introduced in Sect. 2.5. The definition directly implies that for an R-probability
P , L(∅) = 0 and U(Ω) = 1 through the non-emptiness of the credal set.
An R-probability is directly connected to a set of precise probabilities via its
credal set. It is also apparent that for any set of precise probabilities, we may
construct an R-probability s.t. this set will be a subset of its credal set, but such
construction may not be unique. To specify an R-probability, we would need to
define the L and R functions for all the elements in the respective algebra A, which
is quite impractical.
Simpler IP models can be used to assess judgements about more complex ones.
If we specify an R-probability via a (possibly finite) set of precise models, we
automatically include all their convex combinations in the structure of such an
R-probability. Given two K-probabilities p, q ∈ M, the credal set of some Rprobability, for all their convex combinations r = λp + (1 − λ)q, λ ∈ [0, 1],
r(E) will take values in between p(E), q(E) for all the events E ∈ A due to the
axioms of probability measures. Thus, L(E) ≤ r(E) ≤ U(E), so r ∈ M.
Therefore, the credal set of an R-probability is equal to its convex hull.
A one-to-one correspondence between probability bounds and the underlying
credal set is a desired feature in IP theory. In Weichselberger’s treatment, this may
be achieved by tightening the requirements on the probability bounds L, U .
Definition 2.7 An R-probability P , which also satisfies
∀E ∈ A :
inf
p∈M
p(E) = L(E)
∧
sup
p∈M
p(E) = U(E)
,
is called an F-probability and the corresponding tuple F := (Ω, A, L, U),
the F-probability field.
The letter F- indicates feasible. It corresponds to the notion of coherence
introduced in Sect. 2.5. For every F-probability, we also immediately imply that
L(Ω) = 1 and U(∅) = 0, as this is true for each of the elements of the credal set,
therefore also their infimum and supremum. The axioms of F-probabilities directly
imply a relation which is reoccurring throughout many parts of IP theory and which
53
Definition 2.6 (continued)
We will call the tuple R := (Ω, A, L, U) a R-probability field. The set M
from the second axiom is called the structure (Weichselberger) or the credal
set (Walley) of R.
The letter R- indicates reasonable. It corresponds to the property of avoiding sure
loss introduced in Sect. 2.5. The definition directly implies that for an R-probability
P , L(∅) = 0 and U(Ω) = 1 through the non-emptiness of the credal set.
An R-probability is directly connected to a set of precise probabilities via its
credal set. It is also apparent that for any set of precise probabilities, we may
construct an R-probability s.t. this set will be a subset of its credal set, but such
construction may not be unique. To specify an R-probability, we would need to
define the L and R functions for all the elements in the respective algebra A, which
is quite impractical.
Simpler IP models can be used to assess judgements about more complex ones.
If we specify an R-probability via a (possibly finite) set of precise models, we
automatically include all their convex combinations in the structure of such an
R-probability. Given two K-probabilities p, q ∈ M, the credal set of some Rprobability, for all their convex combinations r = λp + (1 − λ)q, λ ∈ [0, 1],
r(E) will take values in between p(E), q(E) for all the events E ∈ A due to the
axioms of probability measures. Thus, L(E) ≤ r(E) ≤ U(E), so r ∈ M.
Therefore, the credal set of an R-probability is equal to its convex hull.
A one-to-one correspondence between probability bounds and the underlying
credal set is a desired feature in IP theory. In Weichselberger’s treatment, this may
be achieved by tightening the requirements on the probability bounds L, U .
Definition 2.7 An R-probability P , which also satisfies
∀E ∈ A :
inf
p∈M
p(E) = L(E)
∧
sup
p∈M
p(E) = U(E)
,
is called an F-probability and the corresponding tuple F := (Ω, A, L, U),
the F-probability field.
The letter F- indicates feasible. It corresponds to the notion of coherence
introduced in Sect. 2.5. For every F-probability, we also immediately imply that
L(Ω) = 1 and U(∅) = 0, as this is true for each of the elements of the credal set,
therefore also their infimum and supremum. The axioms of F-probabilities directly
imply a relation which is reoccurring throughout many parts of IP theory and which