2 Introduction to Imprecise Probabilities
59
2.4.3 Neighbourhood Models
One simple way of defining an IP model is by taking a neighbourhood of some
precise one. In the settings of optimisation under uncertainty, such models would
provide a straightforward way for imposing robustness of the results. Such models
may also be directly used for sensitivity analysis and for the analysis of the
robustness of statistical procedures [25].
There are several ways to construct an IP from a baseline precise distribution
P 0 [3, Ch. 4]. The Pari–Mutuel model enables us to directly evaluate the lower and
upper probabilities it encodes. It originated with the intention to ensure bookmakers
a positive expected gains from a series of lotteries (i.e. buying tickets cheaper than
their known expected gains and selling them for more). For a fixed baseline P 0
and a contamination parameter , the lower and upper probabilities of events are
defined as
P (E) = max{(1 + 0 (E) − , 0},
P (E) = min{(1 + 0 (E), 1}.
Another widely used model is the linear-vacuous model, which is directly
defined for the lower and upper expected values (also for probabilities using
Eq. (2.8)). The idea is to construct a mixture of the baseline model P 0 and a vacuous
model which assigns the infimum and supremum values of a function as their lower
and upper expectations. For a contamination parameter and RV X ∈ L(Ω),
E[X] = (1 − )E P 0 [X] + inf
Ω
X(ω)
From the IP point of view, the lower and upper probabilities induced by the
Pari–Mutuel model are 2-monotone, and those from the linear-vacuous one are
∞-monotone capacities. Both models are therefore coherent. Therefore, for the
Pari–Mutuel, the bounds on expectations can be computed using the Choquet
integral (Theorem 2.5).
2.4.4 Random Sets
A set-valued evidence may be encountered in numerous practical scenarios, may it
be the error bounds of measuring devices or an interval-valued expert elicitation.
There exists an approach for handling these within precise probability theory in
the case that we know that the imprecision is not inherent to the actual realisation
of the experiment and only comes as a coarsening of precise values via our
imperfect methods. In such a case, we may introduce an additional assumption on
the stochastic nature of how the coarsening occurs, a conditional model on where
59
2.4.3 Neighbourhood Models
One simple way of defining an IP model is by taking a neighbourhood of some
precise one. In the settings of optimisation under uncertainty, such models would
provide a straightforward way for imposing robustness of the results. Such models
may also be directly used for sensitivity analysis and for the analysis of the
robustness of statistical procedures [25].
There are several ways to construct an IP from a baseline precise distribution
P 0 [3, Ch. 4]. The Pari–Mutuel model enables us to directly evaluate the lower and
upper probabilities it encodes. It originated with the intention to ensure bookmakers
a positive expected gains from a series of lotteries (i.e. buying tickets cheaper than
their known expected gains and selling them for more). For a fixed baseline P 0
and a contamination parameter , the lower and upper probabilities of events are
defined as
P (E) = max{(1 + 0 (E) − , 0},
P (E) = min{(1 + 0 (E), 1}.
Another widely used model is the linear-vacuous model, which is directly
defined for the lower and upper expected values (also for probabilities using
Eq. (2.8)). The idea is to construct a mixture of the baseline model P 0 and a vacuous
model which assigns the infimum and supremum values of a function as their lower
and upper expectations. For a contamination parameter and RV X ∈ L(Ω),
E[X] = (1 − )E P 0 [X] + inf
Ω
X(ω)
From the IP point of view, the lower and upper probabilities induced by the
Pari–Mutuel model are 2-monotone, and those from the linear-vacuous one are
∞-monotone capacities. Both models are therefore coherent. Therefore, for the
Pari–Mutuel, the bounds on expectations can be computed using the Choquet
integral (Theorem 2.5).
2.4.4 Random Sets
A set-valued evidence may be encountered in numerous practical scenarios, may it
be the error bounds of measuring devices or an interval-valued expert elicitation.
There exists an approach for handling these within precise probability theory in
the case that we know that the imprecision is not inherent to the actual realisation
of the experiment and only comes as a coarsening of precise values via our
imperfect methods. In such a case, we may introduce an additional assumption on
the stochastic nature of how the coarsening occurs, a conditional model on where