Extreme points of the credal sets generated by comparative probabilities

Abstract

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.

A Study of the Set of Probability Measures Compatible with Comparative Judgements

“…Equivalently, the comparative judgements can be represented in terms of a (possibly partial) binary relation on 2 X , the power set of the possibility space X , with A B being equivalent to (A, B) ∈ C . Miranda and Destercke [11] exclusively dealt with comparative assessments that consist of comparative judgements that concern singletons, or equivalently, are a subset of {({x}, {y}): x, y ∈ X }. We follow them in calling such comparative assessments elementary.…”

“…Let Σ X denote the set of all probability mass functions on X . We follow the authors of [8,11,13,17] in considering the set…”

“…Specifically, we shall investigate in which cases this set is non-empty, the number of its extreme points and their features, and the properties of its associated lower probability. While most earlier work on comparative probabilities has mainly focused on the complete case-that is, the case where any two events are compared-ours is not the first study of the incomplete one; in this respect, the most influential works for this paper are those of Walley [17,Section 4.5] and Miranda and Destercke [11]. Our present contribution has the same goals as that of Miranda and Destercke, but our setting is more general: where they exclusively focused on the specific case of comparisons between elementary events, we generalise some of their results to the case of comparisons between arbitrary events.…”

Concepts for decision making under severe uncertainty with partial ordinal and partial cardinal preferences

“…for lower probabilites, cf. Derks and Kuipers (2002) and Wallner (2007)), convenient cases exist where furthermore efficient enumeration procedures are available (such special cases include ordinal probabilities (cf., Kofler (1989, p. 26)), comparative probabilites (cf., Miranda and Destercke (2015)), necessity measures (cf., Schollmeyer (2015)), p-boxes (cf., ), probability intervals (cf., Weichselberger and Pöhlmann (1990, Chapter 2) et al (1994)) or pari-mutuel models (cf., )).…”

Shapley and Banzhaf Values as Probability Transformations

“…It was proven in 41 that the core of these models can be given quite a neat structure, and that it has at most 2 n−1 different extreme points. However, the lower probability induced by this core need not be 2-monotone in general (see 41 , Section 4.3). Taking this into account, it is not surprising to see that the pignistic transformation cannot be computed by means of Eq.…”