Quantitative possibility theory

Quantitative possibility theory has focused on the formal links between numerical possibility and probability. A degree of possibility (resp. necessity) is then an upper (resp. lower) bound of a probability degree. This link enables a unified view of quantitative uncertainty theories to be built in the setting of imprecise probabilities. Formal connections have been established between numerical possibility distributions, belief functions, likelihood functions, as well as alternative simple representations of imprecise probabilities such as p-boxes.

This formal setting paves the way to

  • reinterpreting some basic notions of statistics such as the maximum likelihood principle, prediction and confidence intervals, as well as probabilistic inequalities in terms of possibility measures.
  • extending tools for descriptive statistics to incomplete (interval) data analysis in the setting of random sets. However, sets are attached an epistemic meaning as representing ill-known precise values.
  • developing uncertainty propagation methods where uncertainty is due as much to incomplete knowledge as to the variable nature of phenomena, by mixing probabilities and possibilities, in the setting of imprecise probabilities. It leads to uncertainty management technique that explicitly handle partial ignorance.
  • Finally quantitative possibility theory also offers a general setting for information fusion that encompasses the least square approach as a special case.
  • More recently we have contributed to the theory of belief functions, trying to extend the support functions modeling unreliable testimonies to include prejudices of the receiver so as to interpret all non-dogmatic belief functions as the result of a (perjudiced) fusion process.

Applications of quantitative possibility theory and imprecise probability have been developed in areas such as risk analysis and kriging with incomplete information ( ANR project CRISCO2 on C02 underground storage till 2011), data reconciliation for material flow analysis with application to rare earths based on expert information (ANR project ASTER started in 2012) and fault-tree analysis with applications to aircraft maintenance (in cooperation with Airbus, ISAE et ONERA), supply chains (see also the page on Decision).


  • Felipe Aguirre, Sébastien Destercke, Didier Dubois, Mohamed Sallak, Christelle Jacob. Inclusion/exclusion principle for belief functions. In : International Journal of Approximate Reasoning, Elsevier, Vol. 55 N. 8, pp. 1708-1727, 2014. (pdf)
  • Cédric Baudrit, Inès Couso, Didier Dubois. Joint propagation of probability and possibility in risk analysis: Towards a formal framework. In : International Journal of Approximate Reasoning, Elsevier, Vol. 45, pp. 82-105, 2007.
  • Cédric Baudrit, Didier Dubois, Dominique Guyonnet. Joint Propagation and Exploitation of Probabilistic and Possibilistic Information in Risk Assessment. In : IEEE Transactions on Fuzzy Systems, IEEE : Institute of Electrical and Electronics Engineers, Vol. 14 N. 5, pp. 593-607, October 2006.
  • Inès Couso, Didier Dubois. Statistical reasoning with set-valued information: Ontic vs. epistemic views. In : International Journal of Approximate Reasonning, Elsevier, Special issue Harnessing the information contained in low-quality data sources, Vol. 55 N. 7, pp. 1502-1518, 2014. (pdf)
  • Inés Couso, Didier Dubois. A general framework for maximizing likelihood under incomplete data. International Journal of Approximate Reasoning, 93:238-260, 2018.
  • Inès Couso, Didier Dubois, Luciano Sanchez. Random Sets and Random Fuzzy Sets as Ill-Perceived Random Variables, Springer, SpringerBriefs in Computational Intelligence, 1, August 2014.
  • Didier Dubois, Hélène Fargier, Meissa Ababou, Dominique Guyonnet. A fuzzy constraint-based approach to data reconciliation in material flow analysis. International Journal of General Systems, Vol. 43 N. 8, p. 787-809, 2014.
  • D. Dubois, D. Guyonnet. Risk-informed decision-making in the presence of epistemic uncertainty. In : International Journal of General Systems, Vol. 40 N. 2, p. 145-167, 2011.
  • Didier Dubois, Eyke Hullermeier. Comparing probability measures using possibility theory: A notion of relative peakedness. In : International Journal of Approximate Reasoning, Elsevier, Vol. 45, pp. 364-385, 2007.
  • Didier Dubois, Weiru Liu, Jianbing Ma, Henri Prade. The basic principles of uncertain information fusion. An organised review of merging rules in different representation frameworks. Information Fusion, Vol. 32, p. 12-39, 2016.
  • Didier Dubois, Henri Prade. A set-theoretic view of belief functions. In : Classic works of the Dempster-Shafer theory of belief functions, Logical operations and approximations by fuzzy sets. Ronald Yager, Liping Liu (Eds.), Springer, 14, pp. 375-410, Vol. 219, Studies in fuzziness and soft computing, 2008.
  • Didier Dubois, Henri Prade, Philippe Smets. A definition of subjective possibility. In : International Journal of Approximate Reasoning, Elsevier, Vol. 48, pp. 352-364, 2008. (pdf)
  • Roger Flage, Didier Dubois, Terje Aven. Combined analysis of unique and repetitive events in quantitative risk assessment. International Journal of Approximate Reasoning, Vol. 70, p. 68-78, 2016.
  • Jérôme Fortin, Hélène Fargier, Didier Dubois. Gradual numbers and their application to fuzzy interval analysis. In : IEEE Transactions on Fuzzy Systems, IEEE : Institute of Electrical and Electronics Engineers, Vol. 16 N. 2, pp. 388-402, 2008.
  • Kevin Loquin, Didier Dubois. A fuzzy interval analysis approach to kriging with ill-known variogram and data. In : Soft Computing, Springer-Verlag, Special issue Knowledge extraction from low quality data: theoretical, methodological and practical issues, Vol. 16 N. 5, pp. 769-784, 2012. (pdf)
  • Mathieu Serrurier, Henri Prade. An informational distance for estimating the faithfulness of a possibility distribution, viewed as a family of probability distributions, with respect to data. In : International Journal of Approximate Reasoning, Elsevier, Vol. 54, pp. 919-933, 2013. (pdf)