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D. Krpelík and T. Basu
Example 2.4 Let us assume that we have a collection of open-interval-valued
measurements: {(0.2, 0.6), (0.4, 0.8), (0.1, 0.3)}.
By the Laplace indifference principle, we assign to each of the interval an
equal mass m =
1
3 . With such a mass function, we can construct lower and
upper probabilities via Eqs. (2.12) and (2.13). For example,
L([0.3, 1]) = m((0.4, 0.8)) =
1
3
U([0.3, 1]) = m((0.2, 0.6)) + m((0.4, 0.8)) =
2
3
.
Capacities also provide us means for calculating the bounds not only on
probabilities of events but also on the expected values of RVs. If an F-probability is
viewed as a pair of capacities, we can define an integration functional, which will
enable us to compute the bounds on the expected value, similar to which we are
used to from the precise probability theory (Definition 2.4).
Definition 2.13 For a capacity g : A → R and a real-valued function f
measurable on A, the Choquet integral is defined as
(C)
f dg =
∞
0
g(f ≥ x)dx +
0
−∞
(g(f ≥ x) − 1)dx,
where the integrals on the right-hand side are Riemann’s and f ≥ x denotes
{t ∈ Ω : f (t) ≥ x}.
Theorem 2.5 For a coherent 2-monotone lower probability g : A → R, the
lower expectation of a function f is given by the Choquet integral.
E(f ) = (C)
f dg.
Remark: if the capacity represents a 2-monotone upper probability, the
integration would yield the upper expectation.