# Annotated list of publications

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A behavioural model for linguistic uncertainty

Authors: Peter Walley and Gert de Cooman

Abstract: The paper discusses the problem of modelling linguistic uncertainty, which is the uncertainty produced by statements in natural language. For example, the vague statement Mary is young' produces uncertainty about Mary's age. We concentrate on simple affirmative statements of the type subject-is-predicate', where the predicate satisfies a special condition called monotonicity.  For this case, we model linguistic uncertainty in terms of upper probabilities, which are given a behavioural interpretation as betting rates.  Possibility measures and probability measures are special types of upper probability measure.  We evaluate Zadeh's suggestion that possibility measures should be used to model linguistic uncertainty and the Bayesian claim that probability measures should be used.  Our main conclusion is that, when the predicate is monotonic, possibility measures are appropriate models for linguistic uncertainty.  We also discuss a number of assessment strategies for constructing a numerical model.

Published in:  Information Sciences, 2001, vol. 134, pp. 1-37.

A preprint version similar to the published paper can be downloaded: [ pdf ]

A behavioural model for vague probability assessments

Author: Gert de Cooman

Abstract: I present an hierarchical uncertainty model that is able to represent vague probability assessments, and to make inferences based on them. This model can be given an interpretation in terms of the behaviour of a modeller in the face of uncertainty, and is based on Walley's theory of imprecise probabilities. It is formally closely related to Zadeh's fuzzy probabilities, but it has a different interpretation, and a different calculus. Through rationality (coherence) arguments, the hierarchical model is shown to lead to an imprecise first-order uncertainty model that can be used in decision making, and as a prior in statistical reasoning.

Published in: Fuzzy Sets and Systems, 2005, vol. 154, pp. 305-358.

A preprint version similar to the published paper can be downloaded: [ pdf ]

With discussion: papers by Serafín Moral, Lev Utkin, Romano Scozzafava and Lotfi Zadeh. My rejoinder to their comments is Further thoughts on possibilistic previsions: A rejoinder.

A Daniell-Kolmogorov theorem for supremum preserving upper probabilities

Authors: Hugo J. Janssen, Gert de Cooman and Etienne E. Kerre

Abstract: Possibility measures are interpreted as upper probabilities that are in particular supremum preserving. We define a possibilistic process as a special family of possibilistic variables, and show how its possibility distribution functions can be constructed.  We introduce and study the notions of inner and outer regularity for possibility measures. Using these notions, we prove an analogon for possibilistic processes (and possibility measures) of the well-known probabilistic Daniell-Kolmogorov theorem, in the important special case that the variables assume values in compact spaces, and that the possibility measures involved are regular.

Published in: Fuzzy Sets and Systems, 1999, vol 102, pp. 429-444.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Ample fields

Authors: Gert de Cooman and Etienne E. Kerre

Abstract: We study the notion of an ample or complete field, a special case of the well-known fields and $\sigma$-fields of sets. These collections of sets serve as candidates for the domains of possibility and necessity measures, and are therefore important for the further development of a general fuzzy set and possibility theory. The existence of a one-one relationship between ample fields and atomic complete Boolean lattices is proven. Furthermore, the concept of measurability of general  fuzzy sets with respect to ample fields is explored.

Published in: Simon Stevin, vol. 67, pp. 235-244, 1993.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Ample fields as a basis for possibilistic processes

Authors: Hugo J. Janssen,  Gert de Cooman, Etienne E. Kerre

Abstract: Ample fields play an important role in possibility theory. These fields of subsets of a universe, which are additionally closed under arbitrary unions, act as the natural domains for possibility measures.  A set provided with an ample field is then called an ample space. In this paper we generalise Wang's notions of product ample field and product ample space.  Furtheron we make a topological study of ample spaces and their products, and introduce ample subspaces, extensions and one-point extensions of ample spaces. In this way, the topological groundwork is laid for a mathematical theory of possibilistic processes.

Published in:  Fuzzy Sets and Systems, 2001, vol. 120, pp. 445-458.

A preprint version similar to the published paper can be downloaded: [ pdf ]

A new approach to possibilistic independence

Authors: Gert de Cooman and Etienne E. Kerre

Abstract: The introduction of a notion of independence in possibility theory is a problem of long-standing interest. The definitions that have up to now been given in the literature face some difficulties as far as interpretation is concerned. Also, there are inconsistencies between the definition of independence of measurable sets and possibilistic variables. After a discussion of these definitions and their shortcomings, we suggest a new definition, that is consistent in this respect. Furthermore, we show that in the special case of classical, two-valued possibility our definition has a straightforward and natural interpretation.

Published in:  Proceedings of the Third IEEE International Conference on Fuzzy Systems (FUZZ-IEEE '94, The World Congress on Computational Intelligence) (Orlando, Florida, USA, June 26-29, 1994), pp. 1446-1451.

A preprint version similar to the published paper can be downloaded: [ pdf ]

An expanded version of this conference paper is: Possibility theory III: possibilistic independence

A possibilistic model for behaviour under uncertainty

Authors: Gert de Cooman and Peter Walley

Abstract: In modelling uncertainty, it is common to construct some kind of hierarchical model. Such models arise whenever there is a correct' or ideal' uncertainty model but the modeller is uncertain about what it is. Hierarchical models are widely used in Bayesian inference. Several people have proposed hierarchical models which involve possibility distributions but these models do not have any clear operational meaning. This paper describes a new mathematical model which encompasses the earlier models and also has a simple and general economic interpretation, in terms of betting rates concerning whether or not an economic agent will agree to buy or sell a risky investment for a specified price.  We give a general representation theorem which shows that any consistent model of this kind can be interpreted as a model for second-order uncertainty about the beliefs of a Bayesian economic agent.  We describe how the model can be used to generate first-order upper and lower probabilities and to make statistical inferences and decisions. An application to the analysis of two-person noncooperative games is studied in detail.

Published in: Theory and Decision, 2002, vol. 52, pp. 327-374.

A possibilistic Daniell-Kolmogorov theorem

Authors: Hugo J. Janssen, Gert de Cooman, Etienne E. Kerre

Abstract:  We define a possibilistic process as a special family of possibilistic variables, and show how its possibility distribution functions can be constructed.  We introduce and study the notions of inner and outer regularity for possibility measures. Using these notions, we prove an analogon for possibilistic processes (and possibility measures) of the well-known probabilistic Daniell-Kolmogorov theorem, in the important special case that the variables assume values in compact spaces, and that the possibility measures involved are regular.

Published in:  Proceedings of the Seventh International Fuzzy Systems Association World Congress, IFSA '97 (Prague, June 25-29, 1997), vol. 1, pp. 447-453.

A preprint version similar to the published paper can be downloaded: [ pdf ]

A possibilistic uncertainty model in classical reliability theory

Authors: Gert de Cooman and Bart Cappelle

Abstract:  In this paper, it is argued that a possibilistic uncertainty model can be used to represent linguistic uncertainty about the states of a system and of its components. Furthermore, the basic properties of the application of this model to classical reliability theory are studied. The notion of the possibilistic reliability of a system or a component is defined. Based upon the concept of a binary structure function, the important notion of a possibilistic structure function is introduced. It allows us to calculate the possibilistic reliability of a system in terms of the possibilistic reliabilities of its components.

Published in:  Fuzzy Logic and Intelligent Technologies in Nuclear Science, Proceedings of the First International FLINS Workshop (Mol, Belgium, September 14-16, 1994), pp. 19-25.

A preprint version similar to the published paper can be downloaded: [ pdf ]

A thoroughly revised and updated version of this conference paper is: On modeling possibilistic uncertainty in two-state reliability theory

A possibilistic view on fuzzy control

Authors: Gert de Cooman and Etienne E. Kerre

Abstract: With this conference paper, we mainly intend to serve a didactical aim, in giving brief outline of how the workings of the interior, possibilistic, part of generalized fuzzy controllers can, in our view, be best explained to students. We first indicate how linguistic information can be mathematically represented by possibility measures and/or distributions. We then show how approximate reasoning can be done by properly manipulating these possibility measures. To conclude, we explain how a number of existing fuzzy controllers fit into this possibilistic picture.

Published in:  Proceedings of the Workshop on Automation and Control Engineering in Higher Education (Vienna, Austria, July 5-7, 1995), ed. P. Kopacek and P. Gabko, Vienna University of Technology, Vienna, pp. 79-88, invited.

A preprint version similar to the published paper can be downloaded: [ pdf ]

A random set description of a possibility measure and its natural extension

Authors: Gert de Cooman and Dirk Aeyels

Abstract: The relationship is studied between possibility and necessity measures defined on arbitrary spaces, the theory of imprecise probabilities, and elementary random set theory. It is shown how special random sets can be used to generate normal possibility and necessity measures, as well as their natural extensions. This leads to interesting alternative formulas for the calculation of these natural extensions.

Accepted for publication in:  IEEE Transactions on Systems, Man and Cybernetics, Part A, 2000, vol. 30, pp. 124-130.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Belief models: an order-theoretic investigation

Author: Gert de Cooman

Abstract: I show that there is a common order-theoretic structure underlying many of the models for representing beliefs in the literature. After identifying this structure, and studying it in some detail, I argue that it is useful. On the one hand, it can be used to study the relationships between several models for representing beliefs, and I show in particular that the model based on classical propositional logic can be embedded in that based on the theory of coherent lower previsions. On the other hand, it can be used to generalise the coherentist study of belief dynamics (belief expansion and revision) by using an abstract order-theoretic definition of the belief spaces where the dynamics of expansion and revision take place. Interestingly, many of the existing results for expansion and revision in the context of classical propositional logic can still be proven in this much more abstract setting, and therefore remain valid for many other belief models, such as those based on imprecise probabilities.

Published in: Annals of Mathematics and Artificial Intelligence, 2005, vol. 45, pp. 5--34.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Coherence of rules for defining conditional possibilities

Authors: Peter Walley and Gert de Cooman

Abstract: Possibility measures and conditional possibility measures are given a behavioural interpretation as marginal betting rates against events. Under this interpretation, possibility measures should satisfy two consistency criteria, known as avoiding sure loss" and coherence".  We survey the rules that have been proposed for defining conditional possibilities and investigate which of them satisfy our consistency criteria in two situations of practical interest.  Only two of these rules satisfy the criteria in both cases studied, and the conditional possibilities produced by these rules are highly uninformative.  We introduce a new rule that is more informative and is also coherent in both cases.

Published in:  International Journal of Approximate Reasoning, 1999, vol. 21, pp. 63-107.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Coherence of Dempster's conditioning rule in discrete possibilistic Markov models

Authors: Hugo J. Janssen, Gert de Cooman and Etienne E. Kerre

Abstract: We consider discrete possibilistic systems for which the available information is given by one-step transition possibilities and initial possibilities.  These systems can be represented, or modelled, by a collection of variables satisfying a possibilistic counterpart of the Markov condition. This means that, given the values assumed by a selection of variables, the possibility that a subsequent variable assumes some value only depends on the value taken by the most recent variable of the selection. The one-step transition possibilities are recovered by computing the conditional possibility of any two consecutive variables. Under the behavioural interpretation as marginal betting rates against events these conditional' possibilities and the initial possibilities should satisfy the rationality criteria of avoiding sure loss' and coherence'. We show that this is indeed the case when the conditional possibilities are defined using Dempster's conditioning rule.

Published in: International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2000,  vol. 8, pp. 241-252.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Coherent lower previsions in systems modelling: products and aggregation rules

Authors: Gert de Cooman and Matthias C. M. Troffaes

Abstract: We discuss why coherent lower previsions provide a good uncertainty model for solving generic uncertainty problems involving possibly conflicting expert information. We study various ways of combining expert assessments on different domains, such as natural extension, independent natural extension and the type-I product, as well as on common domains, such as conjunction and disjunction. We provide each of these with a clear interpretation, and we study how they are related. Observing that in combining expert assessments no information is available about the order in which they should be combined, we suggest that the final result should be independent of the order of combination. The rules of combination we study here satisfy this requirement.

Published in: Reliability Engineering and System Safety, 2004, vol. 85, pp. 113-134.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Coherent models for discrete possibilistic systems

Authors: Hugo J. Janssen, Gert de Cooman and Etienne E. Kerre

Abstract:  We consider discrete possibilistic systems for which the available information is given by one-step transition possibilities and initial possibilities.  These systems can be represented by a collection of variables satisfying a possibilistic counterpart of the Markov condition. This means that, given the values assumed by a selection of variables, the possibility that a subsequent variable assumes some value is only dependent on the value taken by the most recent variable of the selection. The one-step transition possibilities are recovered by computing the conditional possibility of any two consecutive variables. Under the behavioural interpretation as marginal betting rates against events these conditional' possibilities and the initial possibilities should satisfy the rationality criteria of avoiding sure loss' and coherence'. We show that this is indeed the case when the conditional possibilities are defined using Dempster's conditioning rule.

Published in:  ISIPTA '99: Proceedings of the First International Symposium on Imprecise Probabilities and Their Applications, eds. G. de Cooman, F. G. Cozman, S. Moral and P. Walley, Imprecise Probabilities Project, Gent, 1999, pp. 189-195.

A preprint version similar to the published paper can be downloaded: [ pdf ]

A revised and updated version of this conference paper is: Coherence of Dempster's conditioning rule in discrete possibilistic Markov models.

Confidence relations and comparative possibility

Author: Gert de Cooman

Abstract: I deal with the order-theoretic characterization of possibility measures. I define the notion of a qualitative possibility ordering, which is a generalization of Dubois' qualitative possibility relations in the following sense: they are defined on arbitrary, not necessarily finite universes, and they allow for incomparability. At the same time, I show that any qualitative possibility ordering is not necessarily determined by its distribution relation, and that in general, special extra conditions must be imposed in order to make sure that it would be.

Published in:  Proceedings of EUFIT '96, volume 1 (Fourth European Congress on Intelligent Techniques and Soft Computing, Aachen, Germany, September 2-5, 1996), invited paper, pp. 539-544.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Confidence relations and ordinal information

Author: Gert de Cooman

Abstract: I define confidence relations on Boolean lattices, which can be interpreted as ordinal representations of uncertainty or information. The set of the confidence relations on a given Boolean lattice can be ordered by set inclusion and thus is shown to form a complete meet-semilattice. We investigate and identify the maximal elements of this structure. Moreover, I prove that it is in particular an algebraic semilattice (or domain), and that its finite elements are precisely the finitely generated confidence relations. I also investigate the relationship with information systems. I define duality and self-duality for confidence relations and show that similar conclusions can be reached if we restrict ourselves to confidence relations which are in particular self-dual. Finally, I discuss the possible incompleteness of confidence relations, and the relation between the abstract mathematical structures studied here, and other existing uncertainty models.

Published in:  Information Sciences, 1997, vol. 104, pp. 241-278.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Confidence relations: an order-theoretic investigation

Author: Gert de Cooman

Abstract: I present a formal study of a special type of relations, which can be interpreted as carriers of ordinal information. I begin this discussion with the definition of confidence relations on a set of events. Then, the set of the confidence relations defined on an event set is provided with a natural partial order relation.  A thorough investigation of the structure of the partially ordered set thus formed leads to a number of interesting conclusions. I show that this structure is an algebraic semilattice, with no greatest element, but containing a set of mutually incomparable maximal elements, which can be interpreted as maxima of ordinal information. Finally, I introduce and study the duality of confidence relations, a notion of central importance in this discussion of ordinal information, which is furthermore intricately linked with absolute certainty.

Published in:  Proceedings of the VI IFSA World Congress, Vol. II, (S� Paulo, Brazil, July 22-28, 1995), pp. 531-534.

A preprint version similar to the published paper can be downloaded: [ pdf ]

A thoroughly revised and expanded version of this conference paper is: Confidence relations and ordinal information

Constructing possibility measures

Authors: Bernard de Baets and Gert de Cooman

Abstract: In this paper, we address some aspects of the extension problem for possibility measures: given the values that a (fuzzy) set mapping takes on a family of (fuzzy) sets, is it possible to extend this mapping to a possibility measure? This problem is shown to be equivalent to a special system of relational equations. When the family of sets considered is a (semi)partition, two important solutions are identified. It is shown that these solutions, and their fuzzifications, play a central part in the treatment of the more general extension problem. This role is shown to be even more conspicuous when the family of fuzzy sets considered is a T-(semi)partition, a notion introduced and studied for the first time in this paper.

Published in:  Proceedings of IIZUMA-NAFIPS '95 (University of Maryland, College Park, MD, VSA, September 17-20, 1995), IEEE Computer Society Press, Los Alamitos, CA, pp. 472-477.

A preprint version similar to the published paper can be downloaded: [ pdf ]

An expanded version of this conference paper is The construction of possibility measures from samples on T-semi-partitions

Describing linguistic information in a behavioural context: possible or not?

Author: Gert de Cooman

Abstract: The paper discusses important aspects of the representation of linguistic information, using imprecise probabilities with a behavioural interpretation. I define linguistic information as the information conveyed by statements in natural language, but restrict myself to simple affirmative statements of the type subject-is-predicate'. Taking the behavioural stance, as it is described in detail in (Walley, 1991), I investigate whether it is possible to give a mathematical model for this kind of information. In particular, I evaluate Zadeh's suggestion (Zadeh, 1978) that we should use possibility measures to this end. I come to the conclusion that, generally speaking, possibility measures are possible models for linguistic information, but that more work should be done in order to evaluate the suggestion that they may be the only ones.

Published in:  Intelligent Systems: A Semiotic Perspective, Proceedings of the 1996 International Multidisciplinary Conference (Gaithersburg, MD, USA, October 20-23, 1996), pp. 141-150.

A preprint version similar to the published paper can be downloaded: [ pdf ]

A thorougly revised and updated version of this conference paper is: A behavioural model for linguistic uncertainty

Dynamic programming for deterministic discrete-time systems with uncertain gain

Authors: Gert de Cooman and Matthias C. M. Troffaes

Abstract: We generalise the optimisation technique of dynamic programming for discrete-time systems with an uncertain gain function. We assume that uncertainty about the gain function is described by an imprecise probability model, which generalises the well-known Bayesian, or precise, models. We compare various optimality criteria that can be associated with such a model, and which coincide in the precise case: maximality, robust optimality and maximinity. We show that (only) for the first two an optimal feedback can be constructed by solving a Bellman-like equation.

Published in: International Journal of Approximate Reasoning, 2005, vol. 39, pp. 257-278.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Epistemic independence in numerical possibility theory

Authors: Enrique Miranda and Gert de Cooman

Abstract: Numerical possibility measures can be interpreted as systems of upper betting rates for events. As such, they have a special part in the unifying behavioural theory of imprecise probabilities, proposed by Walley. On this interpretation, they should arguably satisfy certain rationality, or consistency, requirements, such as avoiding sure loss and coherence. Using a version of Walley's notion of epistemic independence suitable for possibility measures, we study in detail what these rationality requirements tell us about the construction of independent product possibility measures from given marginals, and we obtain necessary and sufficient conditions for a product to satisfy these criteria. In particular, we show that the well-known minimum and product rules for forming independent joint distributions from marginal ones, are only coherent when at least one of these distributions assume just the values zero and one.

Published in: International Journal of Approximate Reasoning, 2003, vol. 32, pp. 23-42.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Evaluation sets and mappings - the order-theoretic aspect of the meaning of properties

Author: Gert de Cooman

Abstract: I study some aspects of the mathematical representation of the solutions of problems encountered in everyday life, called evaluation problems. These problems consist in having to check whether objects in a given universe satisfy one or more properties. First, evaluation problems under a single property are considered. It is argued that (partial-pre)order relations play a central role in this study. Three equivalent ways of representing the order-theoretic aspect of these solutions are presented, the last of which bears a close resemblance to the representations extant in the literature concerning fuzzy set theory. This resemblance is investigated in several examples.After this basic work, the more complicated study of evaluation problems under more than one property is undertaken. It is argued that this approach is necessary in order to investigate the relations between properties and of course the combinations of properties that lead to such logical operations as not, and, or, ... The approach followed for a single property is generalized and the same order-theoretic methods are used to represent the solutions of problems of this kind. Using this approach, the link with fuzzy sets and their set theoretical operations is made through the study of property combinators, truth-functionality and combination functions. This link is studied in several examples.

Published in:  Introduction to the Basic Principles of Fuzzy Set Theory and Some of Its Applications, ed. Etienne E. Kerre, Communication & Cognition, Ghent, 1991, pp. 159-213.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Extension of coherent lower previsions to unbounded random variables

Authors: Matthias C. M. Troffaes and Gert de Cooman

Abstract: We consider the extension of coherent lower previsions from the set of bounded random variables to a larger set. An ad hoc method in the literature consists in approximating an unbounded random variable by a sequence of bounded ones. Its ‘extended’ lower prevision is then defined as the limit of the sequence of lower previsions of its approximations. We identify the random variables for which this limit does not depend on the details of the approximation, and call them previsible. We thus extend a lower prevision to previsible random variables, and we study the properties of this extension.

Published in:  Intelligent Systems for Information Processing: From Representation to Applications, eds. Bernadette Bouchon-Meunier, Laurent Foulloy and Ronald R. Yager, Elsevier Science, Amsterdam, 2003, pp. 277-288.

A preprint version similar to the published paper can be downloaded: [ pdf ]

First results for a mathematical theory of possibilistic Markov processes

Authors: Hugo Janssen, Gert de Cooman and Etienne E. Kerre

Abstract: We provide basic results for the development of a theory of possibilistic Markov processes.  We define and study possibilistic Markov processes and possibilistic Markov chains, and derive a possibilistic analogon of the Chapman-Kolmogorov equation. We also show how possibilistic Markov processes can be constructed using one-step transition possibilities.

Published in:  Proceedings of IPMU '96, volume III (Information Processing and Management of Uncertainty in Knowledge Based Systems, Granada, Spain, July 1-5, 1996), pp. 1425-1431, invited.

A preprint version similar to the published paper can be downloaded: [ pdf ]

First results for a mathematical theory of possibilistic processes

Authors: Hugo J. Janssen, Gert de Cooman and Etienne E. Kerre

Abstract: This paper provides the measure theoretic basis for a theory of possibilistic processes.  We generalize the definition of a product ample field to an indexed family of ample fields, without imposing an ordering on the index set.  We also introduce the notion 'measurable cylinder' and show that any product ample field can be generated by its associated field of measurable cylinders.  Furthermore, we introduce and study the notions 'ample subspace', 'extension of an ample space' and 'one-point extension of an ample space'. Using these notions, we prove that for any family of possibility distributions pS (where S is a nonempty subset of T) satisfying a natural consistency condition, a family (ft | t in T) of possibilistic variables can be constructed such that the product of the mappings (ft | t in S) has pS as a possibility distribution. As a special case we obtain a possibilistic analogon of the probabilistic Daniell-Kolmogorov theorem, a cornerstone for the theory of stochastic processes.

Published in:  Cybernetics and Systems '96, volume 1 (Proceedings of the 13th European Meeting on Cybernetics and Systems Research, Vienna, Austria, April 9-12, 1996), pp. 341-346.

A preprint version similar to the published paper can be downloaded: [ pdf ]

FLINS-related activities in Russia

Authors: Gert de Cooman, Da Ruan and Alexander Ryjov

Abstract: For FLINS'94, the first international workshop on fuzzy logic and intelligent technologies in nuclear science, held in September 1994 in Mol, Belgium, a total of 35 percent of all accepted papers were submitted by Russian scientists. They were presented by only five participants from Russia. This was due to the limited funding available for the workshop. As a result, some important results from our Russian colleagues with possible applications in the framework of FLINS were not reported on. In this paper, we fill this gap, by summarizing all the contributions (20 extended abstracts in the FLINS'94 proceedings) from Russia to FLINS'94, as a survey of the current FLINS-related activities in Russia.

Published in:  Fuzzy Sets and Systems, 1995, vol. 74, pp. 163-173.

A preprint version similar to the published paper can be downloaded: [ pdf ]

From possibilistic information to Kleene's strong multi-valued logics

Author: Gert de Cooman

Abstract: Possibilistic logic in general investigates how possibilistic uncertainty about propositions is propagated when making inferences in a formal logical system. In this paper, we look at a very particular aspect of possibilistic logic: we investigate how, under certain independence assumptions, the introduction of possibilistic uncertainty in classical propositional logic leads to the consideration of special classes of multi-valued logics, with a proper set of truth values and logical functions combining them. First, we show how possibilistic uncertainty about the truth value of a proposition leads to the introduction of possibilistic truth values. Since propositions can be combined into new ones using logical operators, possibilistic uncertainty about the truth values of the original propositions gives rise to possibilistic uncertainty about the truth value of the resulting proposition. Furthermore, we show that in a limited number of special cases there is truth-functionality, i.e. the possibilistic truth value of the resulting proposition is a function of the possibilistic truth values of the original propositions. This leads to the introduction of possibilistic-logical functions, combining possibilistic truth values. Important classes of such functions, the possibilistic extension logics, result directly from this investigation. Finally, the relation between these logics and Kleene's strong multi-valued systems is established.

Published in: Fuzzy Sets, Logics and Reasoning about Knowledge, eds. Didier Dubois, Erich Peter Klement and Henri Prade, Kluwer Academic Publishers, Dordrecht, 1999, pp. 315-323.

A preprint version similar to the published paper can be downloaded: [ pdf ]

This conference paper is intended as a brief summary of the much more detailed account that can be found in Towards a possibilistic logic.

Further thoughts on possibilistic previsions: A rejoinder

Author: Gert de Cooman

Abstract: I present an hierarchical uncertainty model that is able to represent vague probability assessments, and to make inferences based on them. This model can be given an interpretation in terms of the behaviour of a modeller in the face of uncertainty, and is based on Walley's theory of imprecise probabilities. It is formally closely related to Zadeh's fuzzy probabilities, but it has a different interpretation, and a different calculus. Through rationality (coherence) arguments, the hierarchical model is shown to lead to an imprecise first-order uncertainty model that can be used in decision making, and as a prior in statistical reasoning.

Published in: Fuzzy Sets and Systems, 2005, vol. 154, pp. 375-385.

A preprint version similar to the published paper can be downloaded: [ pdf ]

My rejoinder to comments on my paper A behavioural model for vague probability assessments by Serafín Moral, Lev Utkin, Romano Scozzafava and Lotfi Zadeh.

Generalized possibility and necessity measures on fields of sets

Author: Gert de Cooman

Abstract: I give a generalization of possibility and necessity measures: their domains are extended towards fields of sets, and their codomains towards arbitrary complete lattices. In this way, these measures can be associated with L-fuzzy sets, where L is at least a poset. An important inconsistency problem, intricately linked with this association, is solved. It is argued that order lies at the basis of a mathematical description of vagueness and linguistic uncertainty. The results obtained here allow one to mathematically represent and manipulate linguistic uncertainty in the presence of incomparability.

Published in:  Proceedings of the International ICSC Symposium on Fuzzy Logic (ISFL '95) (Zurich, Switserland, May 26-27,1995), ed. N. C. Steele, ICSC Academic Press, Canada, 1995, pp. A91-A98.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Implicator and coimplicator integrals

Authors: Gert de Cooman and Bernard de Baets

Abstract: We introduce and study implicator and coimplicator integrals, and investigate their possible application in defining the possibility and necessity of fuzzy sets. First, the definition and properties of implicators and coimplicators on bounded posets are discussed. Then, in analogy with the theory of seminormed and semiconormed fuzzy integrals, implicator and coimplicator integrals are defined. Next, we study the properties of these dual types of integrals. We uncover an interesting relationship between implicator and coimplicator integrals, and seminormed and semiconormed fuzzy integrals, which could also be called conjunctor and disjunctor integrals. Finally, we show that coimplicator and implicator integrals can be used to extend the domain of possibility measures and necessity measures from sets to fuzzy sets.

Published in:  Proceedings of IPMU '96, volume III (Information Processing and Management of Uncertainty in Knowledge Based Systems, Granada, Spain, Juli 1-5, 1996), pp. 1433-1438, invited.

A preprint version similar to the published paper can be downloaded : [ pdf ]

Integration and conditioning in numerical possibility theory

Author: Gert de Cooman

Abstract: The paper discusses integration and some aspects of conditioning in numerical possibility theory, where possibility measures have the behavioural interpretation of upper probabilities, that is, systems of upper betting rates. In such a context, integration can be used to extend upper probabilities to upper previsions. It is argued that the role of the fuzzy integral in this context is limited, as it can only be used to define a coherent upper prevision if the associated upper probability is 0-1-valued, in which case it moreover coincides with the Choquet integral. These results are valid for arbitrary coherent upper probabilities, and therefore also relevant for possibility theory. It follows from the discussion that in a numerical context, the Choquet integral is better suited than the fuzzy integral for producing coherent upper previsions starting from possibility measures. At the same time, alternative expressions for the Choquet integral associated with a possibility measure are derived. Finally, it is shown that a possibility measure is fully conglomerable and satisfies Walley's regularity axiom for conditioning, ensuring that it can be coherently extended to a conditional possibility measure using both the methods of natural and regular extension.

Published inAnnals of Mathematics and Artificial Intelligence, 2001, vol. 32, pp. 87-123.

Part of the material in this paper is based on the book chapter: Integration in possibility theory.

Integration in possibility theory

Author: Gert de Cooman

Abstract: The paper discusses integration in possibility theory, both in an ordinal and in a numerical (behavioural) context. It is shown that in an ordinal context, the fuzzy integral has in important part in at least three areas: the extension of possibility measures to larger domains, the construction of product measures from marginals and the definition of conditional possibilities. In a numerical (behavioural) context, integration can be used to extend upper probabilities to upper previsions. It is argued that the role of the fuzzy integral in this context is limited, as it can only be used to define a coherent upper prevision if the associated upper probability is 0-1-valued, in which case it moreover coincides with the Choquet integral. These results are valid for arbitrary coherent upper probabilities, and therefore also relevant for possibility theory. It follows from the discussion that in a numerical context, the Choquet integral is better suited than the fuzzy integral for producing coherent upper previsions starting from possibility measures. At the same time, alternative expressions for the Choquet integral associated with a possibility measure are derived, and a number of coherence and regularity results are proven concerning conditional possibilities.

Published in: Fuzzy Measures and Integrals - Theory and Applications, eds. M. Grabisch, T. Murofushi and M. Sugeno, Physica-Verlag, Heidelberg, 2000, pp.124-160.

A preprint version similar to the published paper can be downloaded: [ pdf ]

The part in this work about numerical possibility theory is the starting point for the more detailed paper: Integration and conditioning in numerical possibility theory.

Learning in games using the imprecise Dirichlet model

Authors: Erik Quaeghebeur and Gert de Cooman

Abstract: We propose a new learning model for finite strategic-form two-player games based on fictitious play and Walley's imprecise Dirichlet model (1996, J. Roy. Statistical Society B, vol. 58, pp. 3-57). This model allows the initial beliefs of the players about their opponent's strategy choice to be vacuous or imprecise instead of being precisely defined. A similar generalization can be made as the one proposed by Fudenberg and Kreps (1993, Games Econ. Behav. 5, 320--367) for fictitious play, where assumptions about immediate behavior are replaced with assumptions about asymptotic behavior. We also obtain similar convergence results for this generalization: if there is convergence, it will be to an equilibrium.

Submitted for publication: 2005.

A preprint of this paper is available on request: mail to Gert de Cooman

Lower desirability functions: a convenient imprecise hierarchical uncertainty model

Author: Gert de Cooman

Abstract: I introduce and study a fairly general imprecise second-order uncertainty model, in terms of lower desirability. A modeller's lower desirability for a gamble is defined as her lower probability for the event that a given subject will find the gamble (at least marginally)  desirable. For lower desirability assessments, rationality criteria are introduced that go back to the criteria of avoiding sure loss and coherence in the theory of (first-order) imprecise probabilities. I also introduce a notion of natural extension that allows the least committal coherent extension of lower desirability assessments to larger domains, as well as to a first-order model, which can be used in statistical reasoning and decision making. The main result of the paper is what I call Precision--Imprecision Equivalence: as far as certain behavioural implications of this model are concerned, it does not matter whether the subject's underlying first-order model is assumed to be precise or imprecise.

Published inISIPTA '99: Proceedings of the First International Symposium on Imprecise Probabilities and Their Applications, eds. G. de Cooman, F. G. Cozman, S. Moral and P. Walley, Imprecise Probabilities Project, Ghent, 1999, pp. 111-120.

A preprint version similar to the published paper can be downloaded: [ pdf ]

A thoroughly reworked version of this conference paper is the journal contribution: Precision-imprecision equivalence in a broad class of imprecise hierarchical uncertainty models.

Lower previsions induced by multi-valued mappings

Authors: Enrique Miranda, Gert de Cooman and Inés Couso

Abstract: We discuss how lower previsions induced by multi-valued mappings fit into the framework of the behavioural theory of imprecise probabilities, and show how the notions of coherence and natural extension from that theory can be used to prove and generalise existing results in an elegant and straightforward manner. This provides a clear example for their explanatory and unifying power.

Published in: Journal of Statistical Planning and Inference, 2005, vol. 133, pp. 173-197.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Marginal extension in the theory of coherent lower previsions

Authors: Enrique Miranda and Gert de Cooman

Abstract: We generalise Walley's Marginal Extension Theorem to the case of any finite number of conditional lower previsions. Unlike the procedure of natural extension, our marginal extension always provides the smallest (most conservative) coherent extensions. We show that they can also be calculated as lower envelopes of marginal extensions of conditional linear (precise) previsions. Finally, we use our version of the theorem to study the so-called forward irrelevant product and forward irrelevant natural extension of a number of marginal lower previsions.

Published in: Journal of Intelligent and Fuzzy Systems, 2007, vol. 46, pp. 188-225.

A preprint version similar to the published paper can be downloaded: [ pdf ].

n-Monotone exact functionals

Authors: Gert de Cooman, Matthias C. M. Troffaes and Enrique Miranda

Abstract: We study n-monotone exact functionals, which constitute a generalisation of $n$-monotone set functions. We investigate their relation to the concepts of coherence and natural extension in the behavioural theory of imprecise probabilities, and improve along the way upon a number of results from the literature. Finally, we indicate how many approaches to integration in the literature fall nicely within the framework of the present study of coherent n-monotone exact functionals. This discussion allows us to characterise which types of integrals can be used to calculate the natural extension of a positive bounded charge.

Submitted for publication: 2005.

A preprint of this paper is available on request: mail to Gert de Cooman

n-Monotone lower previsions

Authors: Gert de Cooman, Matthias C. M. Troffaes and Enrique Miranda

Abstract: We study n-monotone lower previsions, which constitute a generalisation of n-monotone lower probabilities. We investigate their relation with the concepts of coherence and natural extension in the behavioural theory of imprecise probabilities, and improve along the way upon a number of results from the literature.

Accepted for publication in : Journal of Intelligent and Fuzzy Systems, 2005.

A preprint version similar to the published paper can be downloaded: [ pdf ].

A significantly expanded version of this paper, dealing with n-monotone exact functionals rather than the special case of n-monotone lower previsions, is n-Monotone exact functionals.

Non-truth-functional order norms

Author: Gert de Cooman

Abstract:  I defend the introduction of triangular (semi)norms and (semi)conorms on bounded partially ordered sets. First, I give a brief survey of the general properties of these binary operators. Then, I work out a number of examples in diverse fields of mathematics, to show that it is indeed useful and natural to generalize the definition of triangular (semi)norms and (semi)conorms from the unit interval towards bounded posets.

Published in:  Proceedings EUFIT '95, Vol. 1, (Third European Congress on Intelligent Techniques and Soft Computing, Aachen, Germany, August 29-31, 1995), pp. 126-130, invited.

A preprint version similar to the published paper can be downloaded: [ pdf ].

On modeling possibilistic uncertainty in two-state reliability theory

Author: Gert de Cooman

Abstract: I show how a possibilistic uncertainty model can be used to represent and manipulate uncertainty about the states of a system and of its components. At the same time, I present a thorough study of the incorporation of this possibilistic uncertainty model in classical, two-state reliability theory. The possibilistic reliability of a component or system is introduced and studied. Furthermore, I introduce the important notion of a possibilistic structure function, based upon the concept of a classical, two-valued structure function. Under certain conditions of possibilistic independence, it allows the calculation of the possibilistic reliability of a system in terms of the possibilistic reliabilities of its components. Finally, I give straightforward methods for determining a possibilistic structure function from its classical, two-valued counterpart. In this way, I intend to show that a possibilistic uncertainty model in two-state reliability theory is formally analogous to, and certainly not more complicated than, a probabilistic uncertainty model.

Published in:  Fuzzy Sets and Systems, 1996, vol. 83, pp. 215-238.

A preprint version similar to the published paper can be downloaded: [ pdf ]

On the coherence of supremum preserving upper previsions

Authors: Gert de Cooman and Dirk Aeyels

Abstract: We study certain aspects of the relation between possibility measures and the theory of imprecise probabilities. It is shown that a possibility measure is a coherent upper probability iff it is normal. We also prove that a possibility measure is the restriction to events of the natural extension of a special kind of upper probability, defined on a class of nested sets. Next, we go from upper probabilities to upper previsions. We show that if a coherent upper prevision defined on the convex cone of all positive gambles is supremum preserving, then it must take the form of a Shilkret integral associated with a possibility measure. But at the same time, we show that a supremum preserving upper prevision is not necessarily coherent! This makes us look for alternative extensions of possibility measures that are not necessarily supremum preserving, through natural extension.

Published in:  Proceedings of IPMU '96, volume III (Information Processing and Management of Uncertainty in Knowledge Based Systems, Granada, Spain, July 1-5, 1996), pp. 1405-1410, invited.

A preprint version similar to the published paper can be downloaded: [ pdf ]

On the extension of P-consistent mappings

Authors: Lars Boyen, Gert de Cooman and Etienne E. Kerre

Abstract: In this paper, the notion of P-consistency is extended to mappings valued on a complete lattice. It is proven that a P-consistent mapping possesses a distribution if and only if it is extendable to a possibility measure defined on an ample field. A necessary and sufficient condition is given for extendability, and it is shown by counterexamples that this condition is not always satisfied. Finally, sufficient conditions are given under which a P-consistent mapping is always extendable, and it is shown that every complete lattice can be embedded in another complete lattice in such a way that every P-consistent mapping is extendable to a possibility measure taking values in the second complete lattice.

Published in:  Foundations and Applications of Possibility Theory - Proceedings of FAPT '95 (International Workshop on the Foundations and Applications of Possibility Theory, Ghent, Belgium, December 13-15, 1995), eds. Gert de Cooman, Da Ruan and Etienne E. Kerre, pp. 88-98.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Order norms on bounded partially ordered sets

Authors: Gert de Cooman and Etienne E. Kerre

Abstract: We extend the domains of affirmation and negation operators, and more importantly, of triangular (semi)norms and (semi)conorms from the unit interval to bounded partially ordered sets. The fundamental properties of the original operators are proven to be conserved under this extension. This clearly shows that they are essentially based upon order-theoretic notions. Consequently, a rather general order-theoretic invariance study of these operators is undertaken. Also, in a brief algebraic excursion, the notion of weak invertibility of these operators is introduced, and the relation with the order-theoretic concept of residuals is studied. The importance of these results for fuzzy set theory and possibility theory is briefly discussed.

Published in: The Journal of Fuzzy Mathematics, vol. 2, pp. 281-310, 1993.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Possibilistic previsions

Author: Gert de Cooman

Abstract: The paper deals with a possibilistic imprecise second-order probability model. It is argued that such models appear naturally in a number of situations. They lead to the introduction of a new type of previsions, called possibilistic previsions, which at least formally generalise coherent upper and lower previsions. The converse problem is also looked at: given a possibilistic prevision, under what conditions can it be generated by a second-order possibility distribution? This leads to the definition of normality, representability and natural extension of possibilistic previsions. Finally, some attention is paid to the special class of  full possibilistic previsions, which can be formally related to Zadeh's fuzzy probabilities. The results have immediate applicability in decision making and statistical reasoning.

Published in: Proceedings of IPMU'98 (Seventh Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, July 6 - 10, 1998, Paris, France), volume I, Editions E.D.K., Paris, pp. 2-9.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Possibilistic second-order probability models

Author: Gert de Cooman

Abstract: The paper deals with a special type of imprecise second-order probability model, which is possibilistic in nature. It is argued that such models appear naturally in a number of situations. They lead to the introduction of a new type of previsions, called possibilistic previsions, which at least formally generalise coherent upper and lower previsions. The converse problem is also looked at: given a possibilistic prevision, under what conditions can it be generated by a second-order possibility distribution? This leads to the introduction of such notions as normality, representability and natural extension of possibilistic previsions. Finally, some attention is paid to the special class of full possibilistic previsions.

Published in: Advances in Cybernetic Modelling of Complex Systems (Part 5 of Proceedings of InterSymp ?97, Baden-Baden, Germany, 18-23 August 1997),  ed. G. E. Lasker, pp. 6-10, invited.

This is a preliminary version of a more detailed conference paper, called Possibilistic previsions.

Possibility and necessity integrals

Authors: Gert de Cooman and Etienne E. Kerre

Abstract: We introduce seminormed and semiconormed fuzzy integrals associated with confidence measures. These confidence measures have a field of sets as their domain, and a complete lattice as their codomain. In introducing these integrals, the analogy with the classical introduction of Legesgue integrals is explored and exploited. It is amongst other things shown that our integrals are the most general integrals that satisfy a number of natural basic properties. In this way, our dual classes of fuzzy integrals constitute a significant generalization of Sugeno's fuzzy integrals. A large number of important general properties of these integrals is studied. Furthermore, and most importantly, the combination of seminormed fuzzy integrals and possibility measures on the one hand, and semiconormed fuzzy integrals and necessity measures on the other hand, is extensively looked into. It is shown that these combinations are very natural, and have properties which are analogous to the combination of Lebesgue integrals and classical measures. Using these results, the very basis is laid for a unifying measure- and integral-theoretic account of possibility and necessity theory, in very much the same way as the theory of Lebesgue integration provides a proper framework for a unifying and formal account of probability theory.

Published in:  Fuzzy Sets and Systems, 1996, vol. 77, pp. 207-227.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Possibility measures and possibility integrals defined on a complete lattice

Authors: Gert de Cooman, Guangquan Zhang and Etienne E. Kerre

Abstract:  We consider the definition of possibility measures on complete lattices rather than on complete Boolean algebras of sets. We give a necessary and sufficient condition for the extendability of any mapping to such a possibility measure. We also associate two types of integrals with these possibility measures, and discuss some of their more important properties, amongst which a monotone convergence theorem.

Published in: Fuzzy Sets and Systems, 2001, vol. 120, pp. 459-467.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Possibility measures, random sets and natural extension

Authors: Gert de Cooman and Dirk Aeyels

Abstract: We study the relationship between possibility and necessity measures defined on arbitrary spaces, the theory of imprecise probabilities, and elementary random set theory. We show how special random sets can be used to generate normal possibility and necessity measures, as well as their natural extensions. This leads to interesting alternative formulas for the calculation of these natural extensions.

Published in:  Proceedings of SC '96 (International Workshop on Soft Computing, Kazan, Tatarstan, Russia, October 3-6, 1996).

A preprint version similar to the published paper can be downloaded: [ pdf ]

A revised and updated version of this conference paper is: A random set description of a possibility measure and its natural extension

Possibility theory I: the measure- and integral-theoretic groundwork

Author: Gert de Cooman

Abstract: I provide the basis for a measure- and integral-theoretic formulation of possibility theory. It is shown that, using a general definition of possibility measures, and a generalization of Sugeno's fuzzy integral - the seminormed fuzzy integral, or possibility integral -, a unified and consistent account can be given of many of the possibilistic results extant in the literature. The striking formal analogy between this treatment of possibility theory, using possibility integrals, and Kolmogorov's measure-theoretic formulation of probability theory, using Lebesgue integrals, is explored and exploited. I introduce and study possibilistic and fuzzy variables as possibilistic counterparts of stochastic and real stochastic variables respectively, and develop the notion of a possibility distribution for these variables. The almost everywhere equality and dominance of fuzzy variables is defined and studied. The proof is given for a Radon-Nikodym-like theorem in possibility theory. Following the example set by the classical theory of integration, product possibility measures and multiple possibility integrals are introduced, and a Fubini-like theorem is proven. In this way, the groundwork is laid for a unifying measure- and integral-theoretic treatment of conditional possibility and possibilistic independence, discussed in more detail in Part II and Part III of this series of three papers.

Published in:  International Journal of General Systems, 1997, vol. 25, pp. 291-323.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Possibility theory II: conditional possibility

Author: Gert de Cooman

Abstract: It is shown that the notion of conditional possibility can be consistently introduced in possibility theory, in very much the same way as conditional expectations and probabilities are defined in the measure- and integral-theoretic treatment of probability theory. I write down possibilistic integral equations which are formal counterparts of the integral equations used to define conditional expectations and probabilities, and use their solutions to define conditional possibilities. In all, three types of conditional possibilities, with special cases, are introduced and studied. I explain why, like conditional expectations, conditional possibilities are not uniquely defined, but can only be determined up to almost everywhere equality, and I assess the consequences of this nondeterminacy. I also show that this approach solves a number of consistency problems, extant in the literature.

Published in:  International Journal of General Systems, 1997, vol. 25, pp. 325-351.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Possibility theory III: possibilistic independence

Author: Gert de Cooman

Abstract: The introduction of the notion of independence in possibility theory is a problem of long-standing interest. Many of the measure-theoretic definitions that have up to now been given in the literature face some difficulties as far as interpretation is concerned. Also, there are inconsistencies between the definition of independence of measurable sets and possibilistic variables. After a discussion of these definitions and their shortcomings, a new measure-theoretic definition is suggested, which is consistent in this respect, and which is a formal counterpart of the definition of stochastic independence in probability theory. In discussing the properties of possibilistic independence, I draw from the measure- and integral-theoretic treatment of possibility theory, discussed in Part I of this series of three papers.  I also investigate the relationship between this definition of possibilistic independence and the definition of conditional possibility, discussed in detail in Part II of this series. Furthermore, I show that in the special case of classical, two-valued possibility the definition given here has a straightforward and natural interpretation.

Published in: International Journal of General Systems, 1997, vol. 25, pp. 353-371.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Practical implementation of possibilistic probability mass functions

Authors: Leen Gilbert, Gert de Cooman and Etienne E. Kerre

Abstract: Probability assessments of events are often linguistic in nature. We model them by means of possibilistic probabilities (a version of Zadeh's fuzzy probabilities with a behavioural interpretation) with a suitable shape for practical implementation (on a computer). Employing the tools of interval analysis and the theory of imprecise probabilities we argue that the verification of coherence for these possibilistic probabilities, the corrections of non-coherent to coherent possibilistic probabilities and their extension to other events and gambles can be performed by finite and exact algorithms. The model can furthermore be transformed into an imprecise first-order model, useful for decision making and statistical inference.

Published in: Proceedings of the Fifth Workshop on Uncertainty Processing (WUPES 2000, Jindrichuv Hradec, Czech republic, June 21-24, 2000), pp. 90-101.

A preprint version similar to the published paper can be downloaded: [ pdf ]

An expanded journal version of this conference paper is: Practical implementation of possibilistic probability mass functions.

Practical implementation of possibilistic probability mass functions

Authors: Leen Gilbert, Gert de Cooman and Etienne E. Kerre

Abstract: Probability assessments of events are often linguistic in nature. We model them by means of possibilistic probabilities (a version of Zadeh's fuzzy probabilities with a behavioural interpretation) with a suitable shape for practical implementation (on a computer). Employing the tools of interval analysis and the theory of imprecise probabilities we argue that the verification of coherence for these possibilistic probabilities, the corrections of non-coherent to coherent possibilistic probabilities and their extension to other events and gambles can be performed by finite and exact algorithms. The model can furthermore be transformed into an imprecise first-order model, useful for decision making and statistical inference.

Published in: Soft Computing, 2003, vol. 7, pp. 304-309.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Precision-imprecision equivalence in a broad class of imprecise hierarchical uncertainty models

Author: Gert de Cooman

Abstract: Hierarchical models are rather common in uncertainty theory. They arise when there is a correct' or `ideal' (so-called first-order) uncertainty model about a phenomenon of interest, but the modeller is uncertain about what it is.   The modeller's uncertainty is then called second-order uncertainty. For most of the hierarchical models in the literature, both the first and the second-order models are precise, i.e., they are based on classical probabilities. In the present paper, I propose a specific hierarchical model that is imprecise at the second level, which means that at this level, lower probabilities are used. No restrictions are imposed on the underlying first-order model: that is allowed to be either precise or imprecise. I argue that this type of hierarchical model generalises and includes a number of existing uncertainty models, such as imprecise probabilities, Bayesian models, and fuzzy probabilities. The main result of the paper is what I call Precision--Imprecision Equivalence: the implications of the model for decision making and statistical reasoning are the same, whether the underlying first-order model is assumed to be precise or imprecise.

Published in: Journal of Statistical Planning and Inference, 2002, vol. 105, pp. 175-198.

A preprint version similar to the published paper can be dowloaded: [ pdf ]

Some remarks on stationary possibilistic processes

Authors: Hugo Janssen, Gert de Cooman and Etienne E. Kerre

Abstract:  We investigate the following extendability problem for systems, for which the available information is given by a monotone set mapping on the field of measurable cylinders of a product ample space: given that this set mapping is invariant under a measurable transformation of that space, is it possible to find invariant monotone extensions of the set mapping to all sets of the ample space? We first show that the outer and inner measures of the set mapping always have the desired invariance property.  If the system that we are dealing with is possibilistic, a number of sufficient conditions are derived to ensure the invariance of the greatest possibilistic extension of the set mapping.  Consequently stationary possibilistic processes can be represented by a shift-invariant possibility measure on their basic space.  As an illustration for our results, we show that possibilistic Markov processes with stationary transition possibilities and stationary initial possibilities are stationary processes.

Published in: Fuzzy Logic and Intelligent Technology for Nuclear Science and Industry (Proceedings of the Third International FLINS Workshop, Antwerp,  Belgium, 14-16 September 1998), eds. D. Ruan, H. A. Abderrahim, P. D'hondt and E. E. Kerre, World Scientific, Singapore, 1998, pp. 52-60.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Supremum preserving upper probabilities

Authors: Gert de Cooman and Dirk Aeyels

Abstract: We study the relation between possibility measures and the theory of imprecise probabilities, and argue that possibility measures have an important part in this theory. It is shown that a possibility measure is a coherent upper probability if and only if it is normal. A detailed comparison is given between the possibilistic and natural extension of an upper probability, both in the general case and for upper probabilities defined on a class of nested sets. We prove in particular that a possibility measure is the restriction to events of the natural extension of a special kind of upper probability, defined on a class of nested sets. We show that possibilistic extension can be interpreted in terms of natural extension. We also prove that when either the upper or the lower distribution function of a random quantity is specified, possibility measures very naturally emerge as the corresponding natural extensions. Next, we go from upper probabilities to upper previsions. We show that if a coherent upper prevision defined on the convex cone of all nonnegative gambles is supremum preserving, then it must take the form of a Shilkret integral associated with a possibility measure. But at the same time, we show that such a supremum preserving upper prevision is never coherent unless it is the vacuous upper prevision with respect to a nonempty subset of the universe of discourse.

Published in: Information Sciences, 1999, vol. 118, pp. 173-212.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Symmetry of models versus models of symmetry

Authors: Gert de Cooman and Enrique Miranda

Abstract: A model for a subject beliefs about a certain phenomenon may exhibit symmetry, in the sense that it is invariant under certain transformations. On the other hand, such a model may be intended to represent that the subject believes or knows that the phenomenon under study exhibits symmetry. We defend the view that these are fundamentally different things, even though the difference cannot be captured by Bayesian belief models. In fact, the failure to distinguish between both situations leads to Laplace's so-called Principle of Insufficient Reason, which has been criticised extensively in the literature, and which led to the rejection of Bayesian methods in the nineteenth and early twentieth century, in favour of frequentist approaches to probability.
We show that there are belief models (imprecise probability models, coherent lower previsions) that generalise and include the more traditional Bayesian models, but where this fundamental difference can be captured. This leads to two notions of symmetry for such models: weak invariance (representing symmetry of beliefs) and strong invariance (modelling beliefs of symmetry). We discuss various mathematical as well as more philosophical aspects of these notions. We also discuss a few examples to show the relevance of our findings both to probabilistic modelling and to statistical inference.

Published in: Probability and Inference: Essays in Honor of Henry E. Kyburg, Jr., eds. William Harper and Gregory Wheeler, pp. 67-149, King's College Publications, London, 2007.

A preprint version similar to the published paper, but with different numbering of theorems and the like, can be downloaded: [ pdf ]

The construction of possibility measures from samples on T-semi-partitions

Authors: Bernard de Baets, Gert de Cooman and Etienne E. Kerre

Abstract: We address the (generalized) extension problem for possibility measures: given a map defined on a family of (fuzzy) sets, is it possible to extend it to a (generalized) possibility measure? The extension problem for possibility measures is known to be equivalent to a system of sup-T equations, with T a t-norm. A key role is played by the greatest solution (of type inf-I, with I a border implicator). When the family of sets considered is a semi-partition, another important solution (of type sup-T, with T a t-norm) can be identified. In the treatment of the generalized possibilistic extension problem, we show that a fuzzification of the greatest solution also plays a central role. On the other hand, an immediate fuzzification of the sup-T type solution is investigated. General necessary and sufficient conditions for this fuzzification to be a solution are established. This fuzzification is then further discussed in the case of a T-semi-partition or a T-partition. Finally, we investigate possible criteria for extendability, inspired by Wang's classical criterion of P-consistency.

Published in:  Information Sciences, special issue: Using fuzzy algebraic structures in intelligent systems, 1998, vol. 106, pp. 3-24.

A preprint version similar to the published paper can be downloaded: [ pdf ]

The formal analogy between possibility and probability theory

Authors: Gert de Cooman

Abstract: It is well known that the theory of probability can be treated and developed in a consistent and uniform way using the classical theory of measure and integration. Indeed, the Russian scientist Kolmogorov identified probability with normalized classical measures and used the Lebesgue theory of integration to give a logically consistent and unifying account of probability theory.  In this paper, we indicate how, in an analogous way, a unified and consistent treatment of possibility theory can be given. Using seminormed fuzzy integrals, a generalization of Sugeno's fuzzy integrals ideally suited for working with possibility measures, we discuss how a theory of possibility can be developed along the same formal lines as the theory of probability.

Published in:  Foundations and Applications of Possibility Theory - Proceedings of FAPT '95 (International Workshop on the Foundations and Applications of Possibility Theory, Ghent, Belgium, December 13-15, 1995), eds. Gert de Cooman, Da Ruan and Etienne E. Kerre, pp. 71-87.

A preprint version similar to the published paper can be downloaded: [ pdf ]

This is a summary of the results published in my series of three papers on the measure-theoretic foundations of ordinal possibility theory:
Possibility theory I: the measure- and integral-theoretic groundwork
Possibility theory II: conditional possibility
Possibility theory III: possibilistic independence

The Hausdorff moment problem revisited

Authors: Enrique Miranda, Gert de Cooman and Erik Quaeghebeur

Abstract: We investigate to what extent finitely additive probability measures on the unit interval are determined by their moment sequence, or by their distribution function. We do this by studying the lower envelope of all finitely additive probability measures with a given moment sequence, or with a given distribution function. Our investigation leads to several elegant expressions for this lower envelope, and it allows us to conclude that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions. Moreover, we see that (lower and upper) Riemann--Stieltjes integrals are, to some limited extent, useful in describing finitely additive solutions to the moment problem.

Submitted for publication:  2006.

The use of linguistic terms in database models

Authors: Robert Groenemans, Etienne E. Kerre, Gert de Cooman and E. Van Ranst

Abstract: Classical database systems have been introduced in the late 50's and have proved their usefulness in various domains. However, their incompetence to deal with vague and imprecise information, has lead to new data base designs. On the other hand the use of linguistic terms has also shown its usefulness. The assignment of linguistic terms to phenomena in order to describe the characteristics or properties of objects is very natural. People make such assignments every day. A drawback of most new database designs is that often the natural aspect of making assignment is lost. In this paper we introduce a new database model based on quasi-order relations (reflexive and symmetric). The proposed model describes the mathematical background of the assignment of values to database attributes, using the theory of evaluation problems and sets. The constructed model offers an interesting new approach to the theory of database design in combination with linguistic terms.

Published in:  Proceedings of IPMU '96, volume III (Information Processing and Management of Uncertainty in Knowledge Based Systems, Granada, Spain, July 1-5, 1996), pp. 1295-1300.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Towards a possibilistic logic

Author: Gert de Cooman

Abstract: I investigate how linguistic information can be incorporated into classical propositional logic. First, I show that Zadeh's extension principle can be justified and at the same time generalized by considerations about transformation of possibility measures. Using these results, I show how linguistic uncertainty about the truth value of a proposition leads to the introduction of the notion of a possibilistic truth value. Since propositions can be combined into new ones using logical operators, linguistic uncertainty about the truth values of the original propositions leads to linguistic uncertainty about the truth value of the resulting proposition. Furthermore, I show that in a number of special cases there is truth-functionality, i.e., the possibilistic truth value of the resulting proposition is a function of the possibilistic truth values of the original propositions. This leads to the introduction of possibilistic-logical functions, combining possibilistic truth values. Important classes of such functions, the possibilistic extension logics, directly result from the above-mentioned investigation, and are studied extensively. Finally, the relation between these logics, and Kleene's strong multi-valued systems is established.

Published in: Fuzzy Set Theory and Advanced Mathematical Applications, ed. Da Ruan, Kluwer Academic, Boston, 1995, pp. 89-133.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Updating with incomplete observations

Authors: Gert de Cooman and Marco Zaffalon

Abstract: Currently, there is renewed interest in the problem, raised by Shafer in 1985, of updating probabilities when observations are incomplete (or set-valued). This is a fundamental problem in general, and of particular interest for Bayesian networks. Recently, Grünwald and Halpern have shown that commonly used updating strategies fail in this case, except under very special assumptions. In this paper we propose a new method for updating probabilities with incomplete observations. Our approach is deliberately conservative: we make no assumptions about the so-called incompleteness mechanism that associates complete with incomplete observations. We model our ignorance about this mechanism by a vacuous lower prevision, a tool from the theory of imprecise probabilities, and we use only coherence arguments to turn prior into posterior (updated) probabilities. In general, this new approach to updating produces lower and upper posterior probabilities and previsions (expectations), as well as partially determinate decisions. This is a logical consequence of the existing ignorance about the incompleteness mechanism. As an example, we use the new updating method to properly address the apparent paradox in the ‘Monty Hall’ puzzle. More importantly, we apply it to the problem of classification of new evidence in probabilistic expert systems, where it leads to a new, so-called conservative updating rule. In the special case of Bayesian networks constructed using expert knowledge, we provide an exact algorithm to compare classes based on our updating rule, which has linear-time complexity for a class of networks wider than polytrees. This result is then extended to the more general framework of credal networks, where computations are often much harder than with Bayesian nets. Using an example, we show that our rule appears to provide a solid basis for reliable updating with incomplete observations, when no strong assumptions about the incompleteness mechanism are justified.

Published in: Artificial Intelligence, 2004, vol. 159, pp. 75-125.

A preprint version similar to the published paper can be downloaded: [ pdf ]

Weak and strong laws of large numbers for coherent lower previsions

Authors: Gert de Cooman and Enrique Miranda

Abstract: We prove weak and strong laws of large numbers for coherent lower previsions, where the lower prevision of a random variable is given a behavioural interpretation as a subject's supremum acceptable price for buying it. Our laws are a consequence of the rationality criterion of coherence, and they can be proven under assumptions that are surprisingly weak when compared to the standard formulation of the law in more classical approaches to probability theory. Moreover, our treatment uncovers an interesting connection between the behavioural theory of coherent lower previsions, and Shafer and Vovk's (2001) game-theoretic approach to probability theory.

Submitted for publication: 2005.

A preprint version similar to the published paper is available on request: mail to Gert de Cooman

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