Qualitative Capacities as Imprecise Possibilities

Dubois, Didier; Prade, Henri; Rico, Agnès

doi:10.1007/978-3-642-39091-3_15

Cited by 10 publications

(6 citation statements)

References 18 publications

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“…Important special cases of capacity are possibility and necessity measures. 1 This paper is based on and extends two previous conference papers [11,21].…”

Section: Qualitative Capacities and Möbius Transformsmentioning

confidence: 95%

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Representing qualitative capacities as families of possibility measures

Dubois

Prade

Rico

2015

International Journal of Approximate Reasoning

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This paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite totally ordered scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multiple-criteria decision-making. We show that any capacity or fuzzy measure ranging on a qualitative scale can be viewed both as the lower bound of a set of possibility measures and the upper bound of a set of necessity measures (a situation somewhat similar to the one of quantitative capacities with respect to imprecise probability theory). We show that any capacity is characterized by a nonempty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and the number of them is characteristic of its complexity. An algorithm is provided to compute the minimal set of possibility measures dominating a given capacity. This algorithm relies on the representation of the capacity by means of its qualitative Möbius transform, and the use of selection functions of the corresponding focal sets. We provide the connection between Sugeno integrals and lower possibility measures. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular modal logics and their neighborhood semantics can be described in terms of qualitative Möbius transforms.

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“…Important special cases of capacity are possibility and necessity measures. 1 This paper is based on and extends two previous conference papers [11,21].…”

Section: Qualitative Capacities and Möbius Transformsmentioning

confidence: 95%

“…It can be shown that the property is equivalent to the existence of two necessity measures such that ∀A, γ (A) = max(N 1 (A), N 2 (A)) (see [21] for a detailed proof of this case). For instance, the capacity in Example 1 is 2-adjunctive.…”

Section: ∀A B C Min γ (A) γ (B) γ (C) ≤ Max γ (A ∩ B) γ (B ∩ Cmentioning

confidence: 99%

Representing qualitative capacities as families of possibility measures

Dubois

Prade

Rico

2015

International Journal of Approximate Reasoning

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“…In practice, we can translate this information by m(A 1 ) ≥ · · · ≥ m(A n ), from which it is easy to extract extreme points. Sets of possibility measures have also been recently investigated from a qualitative perspective by Dubois, Prade, and Rico (2013).…”

Section: Possibility Measuresmentioning

confidence: 99%

Extreme points of the credal sets generated by comparative probabilities

Miranda

Destercke

2015

Journal of Mathematical Psychology

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h i g h l i g h t s• We consider comparative probability models on the singletons of a finite space.• We study the set of probability measures compatible with such a model. • We characterize its extreme points by means of a graphical representation.• We investigate the properties of the lower probability induced by this set.• We provide tight bounds on the number of extreme points. a b s t r a c tWhen using convex probability sets (or, equivalently, lower previsions) as uncertainty models, identifying extreme points can help simplifying various computations or the use of some algorithms. In general, sets induced by specific models such as possibility distributions, linear vacuous mixtures or 2-monotone measures may have extreme points easier to compute than generic convex sets. In this paper, we study extreme points of another specific model: comparative probability orderings between the singletons of a finite space. We characterize these extreme points by means of a graphical representation of the comparative model, and use them to study the properties of the lower probability induced by this set. By doing so, we show that 2-monotone capacities are not informative enough to handle this type of comparisons without a loss of information. In addition, we connect comparative probabilities with other uncertainty models, such as imprecise probability masses.

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“…In the qualitative setting, a monotonic set-function can always be viewed as the lower bound of a set of possibility measures (µ(A) = min n i=1 Π i (A)) which comes down to representing epistemic states expressed by µ as a set of possibility distributions over Ω (e.g. corresponding to several agents) [48].…”

Section: Uncertainty Semanticsmentioning

confidence: 99%