# Annotated list of publications

[ A ] [ B ] [ C ] [ D ] [ E ] [ F ] [ G ] [ I ] [ L ] [ N ] [ O ] [ P ] [ S ] [ T ] [ U ] [ W ]

**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 ]

*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 *p _{S }*(where

*S*is a nonempty subset of

*T*)

*satisfying a natural consistency condition, a family (*

*f*|

_{t }*t in T*) of possibilistic variables can be constructed such that the product of the mappings (

*f*|

_{t }*t in S*) has

*p*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.

_{S}*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 in*: *Annals 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 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, 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 ].

*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

*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 ]

*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 ]

*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