2 Introduction to Imprecise Probabilities
55
Definition 2.8 (continued)
• ∀E ∈ A U , p ∈ M : U(E) ≥ p(E),
• L(∅) = U(∅) = 0, L(Ω) = U(Ω) = 1,
then P = (L, U ) is called a partially determinate R-probability.
We will call (A L , A U ) the support of P .
Definition 2.9 Let (Ω, A, L, U) be a partially determinate R-probability
field with support (A L , A U ). If also
• ∀E ∈ A L : L(E) = inf p∈M p(E),
• ∀E ∈ A U : U(E) = sup p∈M p(E),
then P is called partially determinate F-probability.
For partially determinate F-probabilities, there exists a straightforward way of
calculating probability bounds for events outside of their support. Such procedure
is called normal completion in Weichselberger’s and natural extension in Walley’s
treatment. It simply exploits the extremising property of F-probabilities over their
respective credal sets M, thus
∀E ∈ A : L(A) = inf
p∈M
p(A).
(2.11)
Example 2.3 Let us consider a partially determinate F-probability on
Ω = {0, 1, 2}, A L = A U = {{0}, {1}} with L, U on A L given in Table 2.1,
and a credal set M.
For an arbitrary event E ∈ A, we can calculate its lower probability by
solving the optimisation problem
L(E) = min
p∈M
p(E).
Especially, denoting a := P ({0}), b := P ({1}), c := P ({2}),
L({0, 1}) =
min
0.33 ≤ a ≤ 0.67
0.1 ≤ b ≤ 0.17
a + b + c = 1
a + b = 0.67.
The bounds for the rest of the events in A are given in Table 2.1.