A geometric and game-theoretic study of the conjunction of possibility measures

In this paper, we study the conjunction of possibility measures when they are interpreted as coherent upper probabilities, that is, as upper bounds for some set of probability measures. We identify conditions under which the minimum of two possibility measures remains a possibility measure. We provide graphical way to check these conditions, by means of a zero-sum game formulation of the problem. This also gives us a nice way to adjust the initial possibility measures so their minimum is guaranteed to be a possibility measure. Finally, we identify conditions under which the minimum of two possibility measures is a coherent upper probability, or in other words, conditions under which the minimum of two possibility measures is an exact upper bound for the intersection of the credal sets of those two possibility measures

UNIFYING PRACTICAL UNCERTAINTY REPRESENTATIONS: I. GENERALIZED P-BOXES

Pre-print of final version.International audienceThere exist several simple representations of uncertainty that are easier to handle than more general ones. Among them are random sets, possibility distributions, probability intervals, and more recently Ferson's p-boxes and Neumaier's clouds. Both for theoretical and practical considerations, it is very useful to know whether one representation is equivalent to or can be approximated by other ones. In this paper, we define a generalized form of usual p-boxes. These generalized p-boxes have interesting connections with other previously known representations. In particular, we show that they are equivalent to pairs of possibility distributions, and that they are special kinds of random sets. They are also the missing link between p-boxes and clouds, which are the topic of the second part of this study

Extreme points of credal sets generated by 2-alternating capacities

AbstractThe characterization of the extreme points constitutes a crucial issue in the investigation of convex sets of probabilities, not only from a purely theoretical point of view, but also as a tool in the management of imprecise information. In this respect, different authors have found an interesting relation between the extreme points of the class of probability measures dominated by a second order alternating Choquet capacity and the permutations of the elements in the referential. However, they have all restricted their work to the case of a finite referential space. In an infinite setting, some technical complications arise and they have to be carefully treated. In this paper, we extend the mentioned result to the more general case of separable metric spaces. Furthermore, we derive some interesting topological properties about the convex sets of probabilities here investigated. Finally, a closer look to the case of possibility measures is given: for them, we prove that the number of extreme points can be reduced even in the finite case

Représentation et combinaison d'informations incertaines : nouveaux résultats avec applications aux études de sûreté nucléaires

It often happens that the value of some parameters or variables of a system are imperfectly known, either because of the variability of the modelled phenomena, or because the availableinformation is imprecise or incomplete. Classical probability theory is usually used to treat these uncertainties. However, recent years have witnessed the appearance of arguments pointing to the conclusion that classical probabilities are inadequate to handle imprecise or incomplete information. Other frameworks have thus been proposed to address this problem: the three main are probability sets, random sets and possibility theory. There are many open questions concerning uncertainty treatment within these frameworks. More precisely, it is necessary to build bridges between these three frameworks to advance toward a unified handlingof uncertainty. Also, there is a need of practical methods to treat information, as using these framerowks can be computationally costly. In this work, we propose some answers to these two needs for a set of commonly encountered problems. In particular, we focus on the problems of:- Uncertainty representation- Fusion and evluation of multiple source information- Independence modellingThe aim being to give tools (both of theoretical and practical nature) to treat uncertainty. Some tools are then applied to some problems related to nuclear safety issues.Souvent, les valeurs de certains paramètres ou variables d'un système ne sont connues que de façon imparfaite, soit du fait de la variabilité des phénomènes physiques que l'on cherche à représenter,soit parce que l'information dont on dispose est imprécise, incomplète ou pas complètement fiable.Usuellement, cette incertitude est traitée par la théorie classique des probabilités. Cependant, ces dernières années ont vu apparaître des arguments indiquant que les probabilités classiques sont inadéquates lorsqu'il faut représenter l'imprécision présente dans l'information. Des cadres complémentaires aux probabilités classiques ont donc été proposés pour remédier à ce problème : il s'agit, principalement, des ensembles de probabilités, des ensembles aléatoires et des possibilités. Beaucoup de questions concernant le traitement des incertitudes dans ces trois cadres restent ouvertes. En particulier, il est nécessaire d'unifier ces approches et de comprendre les liens existants entre elles, et de proposer des méthodes de traitement permettant d'utiliser ces approches parfois cher en temps de calcul. Dans ce travail, nous nous proposons d'apporter des réponses à ces deux besoins pour une série de problème de traitement de l'incertain rencontré en analyse de sûreté. En particulier, nous nous concentrons sur les problèmes suivants :- Représentation des incertitudes- Fusion/évaluation de données venant de sources multiples- Modélisation de l'indépendanceL'objectif étant de fournir des outils, à la fois théoriques et pratiques, de traitement d'incertitude. Certains de ces outils sont ensuite appliqués à des problèmes rencontrés en sûreté nucléaire