60
D. Krpelík and T. Basu
the actual value lies in the set (e.g. as in the treatment of censored data in reliability
theory Chap. 4). But in some cases, this assumption may be unjustifiable and bias
our assessments. The imprecision may also be caused by the very nature of the of the
experiment, where the random observation itself is set-valued and cannot be treated
using the mentioned method. To rigorously address these situations, probability
theory may be generalised for the set-valued observations into the theory of random
sets [30, 31, 33].
Definition 2.14 Let (Ω, A, P ) be a probability space, S a collection of
subsets of Ω Φ and Φ : Ω → S a map.
If
{ω : Φ(ω) ∩ E = ∅} ∈ A; ∀ compact K ⊂ Ω Φ ,
then we will call Φ a random set.
The definition of a random set is almost identical to that of a RV (Definition 2.2).
We just need to impose proper measurability properties. Nevertheless, the treatment
of the random sets is slightly different. For a random set Φ, we can assess several
claims.
Definition 2.15 Let Φ : Ω → S ⊂ 2 Ω Φ be a random set derived from
probability space (Ω, A, P ). Then, for E ∈ S and x ∈ Ω Φ , we define the
following:
The belief function Bel(E) := P (Φ ⊂ E).
The plausibility function P l(E) := P (Φ ∩ E = ∅).
The contour function C(x) := P (x ∈ Φ).
The belief and plausibility functions corresponds to lower and upper probabilities
and L and R functions from Sect. 2.3, respectively.
Random set theory is a basis for the Dempster–Shafer theory of evidence,
described in Chap. 17 and some modern statistical methods [29]. Random set
models have also been used for sensitivity analysis [34] and uncertainty modelling
in general [22]. As IP models, the belief and plausibility functions induced by
random sets are ∞-monotone capacities and, therefore, coherent lower probabilities.
Therefore, we also know the form for the derived lower and upper expectations via
Theorem 2.5.
Random sets can be used for statistical inference with little assumptions. The
models can (and have been [34]) be constructed from Chebyshev’s inequality
(Theorem 2.6) if only the population mean and variance are known. This represents
D. Krpelík and T. Basu
the actual value lies in the set (e.g. as in the treatment of censored data in reliability
theory Chap. 4). But in some cases, this assumption may be unjustifiable and bias
our assessments. The imprecision may also be caused by the very nature of the of the
experiment, where the random observation itself is set-valued and cannot be treated
using the mentioned method. To rigorously address these situations, probability
theory may be generalised for the set-valued observations into the theory of random
sets [30, 31, 33].
Definition 2.14 Let (Ω, A, P ) be a probability space, S a collection of
subsets of Ω Φ and Φ : Ω → S a map.
If
{ω : Φ(ω) ∩ E = ∅} ∈ A; ∀ compact K ⊂ Ω Φ ,
then we will call Φ a random set.
The definition of a random set is almost identical to that of a RV (Definition 2.2).
We just need to impose proper measurability properties. Nevertheless, the treatment
of the random sets is slightly different. For a random set Φ, we can assess several
claims.
Definition 2.15 Let Φ : Ω → S ⊂ 2 Ω Φ be a random set derived from
probability space (Ω, A, P ). Then, for E ∈ S and x ∈ Ω Φ , we define the
following:
The belief function Bel(E) := P (Φ ⊂ E).
The plausibility function P l(E) := P (Φ ∩ E = ∅).
The contour function C(x) := P (x ∈ Φ).
The belief and plausibility functions corresponds to lower and upper probabilities
and L and R functions from Sect. 2.3, respectively.
Random set theory is a basis for the Dempster–Shafer theory of evidence,
described in Chap. 17 and some modern statistical methods [29]. Random set
models have also been used for sensitivity analysis [34] and uncertainty modelling
in general [22]. As IP models, the belief and plausibility functions induced by
random sets are ∞-monotone capacities and, therefore, coherent lower probabilities.
Therefore, we also know the form for the derived lower and upper expectations via
Theorem 2.5.
Random sets can be used for statistical inference with little assumptions. The
models can (and have been [34]) be constructed from Chebyshev’s inequality
(Theorem 2.6) if only the population mean and variance are known. This represents