Numerical Representation of Uncertainty
Published 1 January 1998
Philippe Smets
Citations16
Generate an AI Snapshot to get a quick, structured summary of this paper.
Study Snapshot
ObjectiveStudy objective
MethodsResearch methodology
PopulationPopulation studied
Sample sizeSample sizes
OutcomesStudy outcomes here
ResultsStudy results comes here
LimitationsResearch study limitations comes here
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
Abstract
Information is perfect when it is consistent, precise and certain. Imperfection can thus be due to inconsistency, imprecision or uncertainty. When perfect information describes a world, it should uniquely determine it.
Keywords
Computer ScienceDecision Sciences
Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference
16,927 Citations1988Judea Pearl
The author provides a coherent explication of probability as a language for reasoning with partial belief and offers a unifying perspective on other AI approaches to uncertainty, such as the Dempster-Shafer formalism, truth maintenance systems, and nonmonotonic logic.
IEEE Transactions on Systems Man and CyberneticsOutline of a New Approach to the Analysis of Complex Systems and Decision Processes
8,667 Citations1973Lotfi A. Zadeh
By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis.
Fuzzy Sets and SystemsFuzzy sets as a basis for a theory of possibility
7,964 Citations1978Lotfi A. Zadeh
The Logic of Scientific Discovery
7,796 Citations2013
Fuzzy Sets and SystemsFuzzy sets as a basis for a theory of possibility
7,280 Citations1999Lotfi A. Zadeh
The theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable.
Mathematics in Science and Engineering/Mathematics in science and engineeringFuzzy Sets and Systems - Theory and Applications
6,430 Citations1980Didier Dubois, Henri Prade
The Annals of Mathematical StatisticsUpper and Lower Probabilities Induced by a Multivalued Mapping
5,972 Citations1967A. P. Dempster
Studies in fuzziness and soft computingUpper and Lower Probabilities Induced by a Multivalued Mapping
5,299 Citations2008Arthur P. Dempster
A distinctive feature of the present approach is a rule for conditioning, or more generally, arule for combining sources of information, as discussed in Sects.
Journal of the American Statistical AssociationOptimal Statistical Decisions.
3,492 Citations1972David Siegmund, Morris H. DeGroot
The MIT Press eBooksReasoning About Knowledge
2,594 Citations1995Ronald Fagin, Joseph Y. Halpern +2 more
Reasoning About Knowledge is the first book to provide a general discussion of approaches to reasoning about knowledge and its applications to distributed systems, artificial intelligence, and game theory.
Artificial IntelligenceThe transferable belief model
2,165 Citations1994Philippe Smets, Robert Kennes
Journal of the Royal Statistical Society Series B (Statistical Methodology)A Generalization of Bayesian Inference
1,971 Citations1968A. P. Dempster
Studies in fuzziness and soft computingA Generalization of Bayesian Inference
1,722 Citations2008Arthur P. Dempster
Procedures of statistical inference are described which generalize Bayesian inference in specific ways and some comments are made on the general class of models which produce upper and lower probabilities, and on the specific models which underlie the suggested inference procedures.
Truth and Probability
1,311 Citations2016Frank Plumpton Ramsey
American Journal of PhysicsProbability, Frequency and Reasonable Expectation
1,293 Citations1946R. T. Cox
Munich Personal RePEc Archive (Ludwig Maximilian University of Munich)The Logic of Decision
1,146 Citations1965Richard C. Jeffrey
Institutional Research Information System (Università degli Studi di Trento)Advances in the Dempster-Shafer theory of evidence
1,021 Citations1994Ronald R. Yager, Janusz Kacprzyk +1 more
The Dempster-Shafer Theory of Evidence is applied as a guide for the management of uncertainty in knowledge-based systems.
Information SciencesA review of fuzzy set aggregation connectives
925 Citations1985Didier Dubois, Henri Prade
An extensive survey on fuzzy set-theoretic operations is provided, and the relevance of the theory of functional equations in the axiomatical construction of classes of such operations and the derivation of functional representations is emphasized.
Fuzzy Sets and SystemsFuzzy sets in approximate reasoning, Part 1: Inference with possibility distributions
808 Citations1991Didier Dubois, Henri Prade
This paper focuses on semantic approaches to approximate reasoning based on fuzzy sets, commonly exemplified by the generalized modus ponens, but also considers applications to current topics in Artificial Intelligence such as default reasoning and qualitative process modeling.
Elsevier eBooksFUZZY MEASURES AND FUZZY INTEGRALS—A SURVEY
764 Citations1993Michio Sugeno
The concept of fuzzy sets (precisely speaking, fuzzy subsets of an ordinary set) is nothing but an extended concept of ordinary sets, and the concept of probabilities is absolutely different from that of sets.
IFAC Proceedings VolumesFuzzy Information and Decision Processes
674 Citations1982Madan M. Gupta
This paper presents a meta-modelling framework for the design of feedback algorithms implementing the experience of a human operator in the feedback loop based upon the fuzzy set theory and possibility theory.
The Logic of Conditionals
673 Citations1975Ernest W. Adams
Journal of the American Statistical AssociationLogic of Statistical Inference.
616 Citations1966A. P. Dempster, Ian Hacking
Cambridge University Press eBooksLogic of Statistical Inference
611 Citations2016Ian Hacking, Jan‐Willem Romeijn
Journal of the Royal Statistical Society Series B (Statistical Methodology)Consistency in Statistical Inference and Decision
561 Citations1961Cedric A. B. Smith
Journal of the American Statistical AssociationThe Evidential Foundations of Probabilistic Reasoning.
517 Citations1995FH, David A. Schum
The British Journal for the Philosophy of ScienceThe Propensity Interpretation of Probability
508 Citations1959Karl R. Popper
The Philosophical ReviewBayes or Bust? A Critical Examination of Bayesian Confirmation Theory.
496 Citations1994David Christensen, John Earman
Revue économiqueDecision, Order and Time in Human Affairs
478 Citations1970Jacques Lécaillon, G. L. S. Shackle +1 more
Machine intelligence and pattern recognitionConstructing the Pignistic Probability Function in a Context of Uncertainty
432 Citations1990Philippe Smets
HAL (Le Centre pour la Communication Scientifique Directe)Non-standard logics for automated reasoning
406 Citations1988Philippe Smets
International Journal of General SystemsA CLASS OF FUZZY MEASURES BASED ON TRIANGULAR NORMS A general framework for the combination of uncertain information
330 Citations1982Didier Dubois, Henri Prade
An axiomatic approach to a broad class of fuzzy measures in the sense of Sugeno is presented via the concept of triangular norm (t-norm for short), which encompasses probability measures, Zadeh's possibility measures and the dual notion of necessity measures.
Fuzzy Sets and SystemsConnectives and quantifiers in fuzzy sets
330 Citations1991Ronald R. Yager
The concept of linguistic quantifiers is introduced and some applications of these structures are looked at, particularly concerned with their application to linguistic summarizers, syllogistic reasoning and multi-criteria decision making.
ACM eBooksSystem <i>Z</i> : A Natural Ordering of Defaults with Tractable Applications to Nonmonotonic Reasoning
325 Citations2022Judea Pearl
The Z-ordering is defined, two extensions are then described, maximum-entropy and conditional entailment, which trade in computational simplicity for semantic refinements.
Possibility Theory: Qualitative and Quantitative Aspects
317 Citations1998Didier Dubois, Henri Prade
Possibility theory offers a simple, non-additive modelling of partial belief, which contrasts with probability theory, and provides a potentially more qualitative treatment ofpartial belief since the operations ‘max’ and ‘min’ play a role somewhat analogous to the sum and the product in probability calculus.
Artificial IntelligenceBayesian and non-bayesian evidential updating
314 Citations1987Henry E. Kyburg
The impact of “uncertain evidence” can be (formally) represented by Dempster conditioning, in Shafer's framework, in the framework of convex sets of classical probabilities by classical conditionalization.
The British Journal for the Philosophy of ScienceTheories of Probability
289 Citations1995Colin Howson
Journal of Mathematical Analysis and Applications⊥-Decomposable measures and integrals for Archimedean t-conorms ⊥
280 Citations1984Siegfried Weber
Artificial IntelligenceEpistemic entrenchment and possibilistic logic
248 Citations1991Didier Dubois, Henri Prade
This note points out the close relationships existing between recent proposals in the theory of belief revision made by Gardenfors based on the notion of epistemic entrenchment, and possibility theory applied to automated reasoning under uncertainty.
arXiv (Cornell University)A New Approach to Updating Beliefs
246 Citations2013Ronald Fagin, Joseph Y. Halpern
It is shown here how to extend the definition of conditional probability to nonmeasurable sets, in order to get notions of inner and outer conditional probabilities, which can be viewed as best approximations to the true conditional probability, given the lack of information.
International Journal of Approximate ReasoningImplication in fuzzy logic
236 Citations1987Philippe Smets, Paul Magrez
Representing Default Rules in Possibilistic Logic
226 Citations1992Salem Benferhat, Didier Dubois +1 more
It is pointed out that the notion of inconsistency tolerant inference in possibilistic logic corresponds to the bold inference in system Z, and how to express defaults by means of qualitative possibility relations is shown.
Artificial IntelligenceOn the justification of Dempster's rule of combination
205 Citations1991Frans Voorbraak
International Journal of Approximate ReasoningReasoning with belief functions: An analysis of compatibility
202 Citations1990Judea Pearl
Assessing their role in evidence pooling, it is found that belief functions offer a rich language for describing the evidence gathered, highly compatible with the way people summarize observations, but the methods available for integrating evidence into a coherent state of belief capable of supporting plausible decisions cannot make use of this richness.
A Mathematical Theory of Hints: An Approach to the Dempster-Shafer Theory of Evidence
183 Citations1995Jürg Kohlas, Paul-André Monney
This chapter discusses the construction of a model of a hint, the role of support, and the structure of support and Plausibility in the general case.
Machine intelligence and pattern recognitionA General Non-Probabilistic Theory of Inductive Reasoning
174 Citations1990Wolfgang Spohn
Probability theory, epistemically interpreted, provides an excellent account of inductive reasoning; however, this does not yet provide a theory of induction—an answer to the question how epistemic states so represented are changed through information or experience.
Studies in fuzziness and soft computingBayesian Updating and Belief Functions
173 Citations2008Jean‐Yves Jaffray
Morgan Kaufmann Publishers Inc. eBooksThe Dempster-Shafer theory of evidence
164 Citations1990Jill Gordon, Edward H. Shortliffe
The advantage of the Dempster-Shafer theory over previous approaches is its ability to model the narrowing of the hypothesis set with the accumulation of evidence, a process that characterizes diagnostic reasoning in medicine and expert reasoning in general.
Lecture notes in computer scienceThe transferable belief model
157 Citations1991Ph. Smets, Yen‐Teh Hsia +4 more
Information SciencesThe degree of belief in a fuzzy event
152 Citations1981Philippe Smets
Epistemic probabilities are better described by belief functions and their definition is extended in order to apply them to fuzzy events.
Journal of Mathematical Analysis and ApplicationsA Semantics for Possibility Theory Based on Likelihoods
150 Citations1997Didier Dubois, Serafı́n Moral +1 more
Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning
145 Citations1991I. R. Goodman, Hung T. Nguyen +1 more
This book discusses thegebraic structure of Conditional Events, an abstraction of the space of conditional events, and the role of Boolean algebras and Lewis' triviality result in its development.
Lecture notes in economics and mathematical systemsA Mathematical Theory of Hints
139 Citations1995Jürg Kohlas, Paul-André Monney
Oxford University Press eBooksNomic Probability and the Foundations of Induction
136 Citations1990John L. Pollock
IEEE Transactions on Systems Man and Cybernetics - Part A Systems and HumansRepresenting partial ignorance
129 Citations1996Didier Dubois, Henri Prade +1 more
It is pointed out that within a numerical framework, two numbers are needed to account for partial ignorance about events, because on top of truth and falsity, the state of total ignorance must be encoded independently of the number of underlying alternatives.
John Wiley & Sons, Inc. eBooksWhat is Dempster-Shafer's model?
124 Citations1994Philippe Smets
This work spells out through examples what are the underlying hypotheses that lead to the selection of an adequate model for a given problem and gives indications on how to choose the appropriate model.
The Philosophical ReviewNomic Probability and the Foundations of Induction.
122 Citations1993Henry E. Kyburg, John L. Pollock
Journal of the Operational Research SocietyConditional Logic in Expert Systems
116 Citations1992J. J. Garnett
European Journal of Operational ResearchFuzzy sets and systems: Theory and applications
113 Citations1983J.F. Baldwin
Conditionalization, Observation, and Change of Preference
108 Citations1976Paul Teller
This chapter discusses the justification of condition (ii) from the point of view of a frequency interpretation of probability or reasonable degree of belief, and discusses the connection between change of belief and change of preference.
International Journal of Approximate ReasoningThe context model: An integrating view of vagueness and uncertainty
90 Citations1993Jörg Gebhardt, Rudolf Kruse
The context model is applied to show a direct comparison of Bayes theory and the Dempster-Shafer theory within the context model, based on the well-known spoiled sandwich effect, the three prisoners problem, and the unreliable alarm paradigm.
Making inconsistency respectable 1: A logical framework for inconsistency in reasoning
82 Citations1992Dov M. Gabbay, Anthony Hunter
Uncertainty in Artificial IntelligenceThe Nature of the unnormalized Beliefs encountered in the Transferable Belief Model
82 Citations1992Philippe Smets
Within the transferable belief model, positive basic belief masses can be allocated to the empty set, leading to unnormalized belief functions, which are analyzed to determine the nature of these un normalized beliefs.
Boston studies in the philosophy of scienceDispositional Probabilities
80 Citations1971James H. Fetzer
The Value of Studying Subjective Evaluations of Probability
79 Citations1974Bruno de Finetti
IEEE Transactions on Systems Man and CyberneticsAxiomatization of qualitative belief structure
78 Citations1991S. K. M. Wong, Yiyu Yao +2 more
The axiomization of qualitative belief provides a foundation to develop a utility theory for decision making based on belief functions and may lead to a better understanding and useful applications of belief structures in approximate reasoning.
Some Connections Between Epistemic Logic and the Theory of Nonadditive Probability
76 Citations1994Philippe Mongin
Lecture notes in computer scienceMaking inconsistency respectable: A logical framework for inconsistency in reasoning, part I — A position paper
75 Citations1991Dov M. Gabbay, Anthony Hunter
There is a need to develop a framework in which inconsistency can be viewed according to context, as a vital trigger for actions, for learning, and as an important source of direction in artificial intelligence.
International Journal of Approximate ReasoningEvidence, knowledge, and belief functions
70 Citations1992Didier Dubois, Henri Prade
This article addresses most of the questions raised by Pearl in the 1990 special issue of the International Journal of Approximate Reasoning on belief functions and belief maintenance in artificial intelligence.
International Journal of Approximate ReasoningRejoinders to comments on “perspectives on the theory and practice of belief functions”
62 Citations1992Glenn Shafer
The case for may own constructive interpretation of probability, probability bounds, and beliefs functions is summarized and how this constructive interpretation bears on many of the issues raised by the discussants are explained.
Conditioning, Kinematics, and Exchangeability
58 Citations1988Richard C. Jeffrey
Artificial IntelligenceThe normative representation of quantified beliefs by belief functions
57 Citations1997Philippe Smets
A set of requirements that justify the use of belief functions are presented and the assessment of the validity of these requirements provides a tool for assessing the relative value of normative models of subjective behaviors.
A Theory of Statistical Evidence
55 Citations1976Glenn Shafer
Fuzzy Sets and SystemsProbability of a fuzzy event: An axiomatic approach
53 Citations1982Philippe Smets
Axioms are proposed that could justify the natural definition of the probability of a fuzzy event initially given by Zadeh based on the postulate that the sum of the conditional probability and its complement given any fuzzy event adds to one or on soft independence for orthogonal sets with independent constitutive elements.
Journal of the Operational Research SocietyConditional Inference and Logic for Intelligent Systems
46 Citations1993Charles Leake
Uncertainty in Artificial IntelligenceJeffrey's rule of conditioning generalized to belief functions.
45 Citations1993Philippe Smets
Jeffrey's rule of conditioning is generalized within the framework of the models based on belief functions and it is shown that several forms of Jeffrey's conditionings can be defined that correspond to the geometricalRule of conditioning and to Dempster's rule, respectively.
Fuzzy Sets and SystemsThe measure of the degree of truth and of the grade of membership
27 Citations1988Philippe Smets, Paul Magrez
A method is presented to define the ‘meaning’ of a fuzzy proposition and a fuzzy set through the construction of fuzzy propositions with non-ambiguous intermediate truth values.
The Philosophical QuarterlyBayes or Bust? A Critical Examination of Bayesian Confirmation Theory.
24 Citations1995Paul Castell, John Earman
Lecture notes in computer scienceDecision-making with Belief Functions and pignistic probabilities
24 Citations2005Nic Wilson
Philippe Smets' method of decisionmaking is described, which transforms a Belief Function into a single probability function, the pignistic probabilityfunction, which is equivalent to the standard approach.
Lecture notes in computer scienceProbability of deductibility and belief functions
19 Citations2005Philippe Smets
An interpretation of Dempster-Shafer theory based on the probability of deducibility is presented and two forms of revision (conditioning) that lead to the geometrical rule of conditioning and to DemPster rule ofconditioning are presented.
Artificial IntelligenceBelief functions and default reasoning
16 Citations2000Salem Benferhat, Alessandro Saffiotti +1 more
This work presents a new approach to dealing with default information based on the theory of belief functions, and defines two systems based on epsilon-belief assignments, where values committed to focal elements are either close to 0 or close to 1.
Modelisation de contradictions par extension de la theorie de l'evidence
3 Citations1997Patrick Jaouen
Theorie de l'evidence : modelisation et application pour un sic
3 Citations1997Jean-Philippe Perin
…