Abstract

Numerical possibility distributions can encode special convex families of probability measures. The connection between possibility theory and probability theory is potentially fruitful in the scope of statistical reasoning when uncertainty due to variability of observations should be distinguished from uncertainty due to incomplete information. This paper proposes an overview of numerical possibility theory. Its aim is to show that some notions in statistics are naturally interpreted in the language of this theory. First, probabilistic inequalites (like Chebychev's) offer a natural setting for devising possibility distributions from poor probabilistic information. Moreover, likelihood functions obey the laws of possibility theory when no prior probability is available. Possibility distributions also generalize the notion of confidence or prediction intervals, shedding some light on the role of the mode of asymmetric probability densities in the derivation of maximally informative interval substitutes of probabilistic information. Finally, the simulation of fuzzy sets comes down to selecting a probabilistic representation of a possibility distribution, which coincides with the Shapley value of the corresponding consonant capacity. This selection process is in agreement with Laplace indifference principle and is closely connected with the mean interval of a fuzzy interval. It sheds light on the “defuzzification” process in fuzzy set theory and provides a natural definition of a subjective possibility distribution that sticks to the Bayesian framework of exchangeable bets. Potential applications to risk assessment are pointed out.

Keywords: Possibility theory; Imprecise probability; Confidence intervals; Uncertainty propagation

Article Outline

1. Introduction

2. Basics of possibility theory

2.1. The logical view

2.2. Graded possibility and necessity

2.3. Conditioning in possibility theory

3. Relationship between probability and possibility theories

3.1. Imprecise probability

3.2. Random sets

3.3. Likelihood functions

4. From confidence sets to possibility distributions

4.1. Basic principles for probability–possibility transformations

4.2. Alternative approaches to probability–possibility transforms

4.3. The continuous case

5. Possibility theory and subjective probability

5.1. A generalized Insufficient Reason principle

5.2. A Bayesian approach to subjective possibility

6. Fuzzy intervals and possibilistic expectations

6.1. Possibilistic cumulative distributions

6.2. Possibilistic integrals

6.3. Expectations of fuzzy intervals

6.4. The mean interval and defuzzification

6.5. The variance of a fuzzy interval

7. Uncertainty propagation with possibility distributions

8. Conclusion

Acknowledgements

References