Possibility Theory and Its Applications: Where Do We Stand?

Abstract

This paper provides an overview of possibility theory, emphasizing its historical roots and its recent developments. Possibility theory lies at the crossroads between fuzzy sets, probability and non-monotonic reasoning. Possibility theory can be cast either in an ordinal or in a numerical setting. Qualitative possibility theory is closely related to belief revision theory, and common-sense reasoning with exception-tainted knowledge in Artificial Intelligence. Possibilistic logic provides a rich representation setting, which enables the handling of lower bounds of possibility theory measures, while remaining close to classical logic. Qualitative possibility theory has been axiomatically justified in a decision-theoretic framework in the style of Savage, thus providing a foundation for qualitative decision theory. Quantitative possibility theory is the simplest framework for statistical reasoning with imprecise probabilities. As such it has close connections with random set theory and confidence intervals, and can provide a tool for uncertainty propagation with limited statistical or subjective information.

Dissolved oxygen prediction using a possibility theory based fuzzy neural network

“…This implies that the simplest membership function is triangular, though not necessarily symmetrical. Also, as we demonstrate below, in some probability-possibility frameworks, a triangular membership function corresponds to a uniform probability distribution -the least specific distribution in that any value is equally probable, and hence represents the most uncertainty (Dubois and Prade, 2015;Dubois et al, 2004).…”

“…Multiple frameworks exist to transform a probability distribution to a possibility distribution and vice versa; a comparison of different conceptual approaches is provided in Klir and Parvais (1992), Oussalah (2000), Jaquin (2010), Mauris (2013) and Dubois and Prade (2015). However, a major issue of implementing fuzzy number based methods in hydrology is that there is no consistent, transparent and objective method to convert observations (e.g.…”

Optimising Fuzzy Neural Network Architecture for Dissolved Oxygen Prediction and Risk Analysis

“…The constraints are such that to find the upper and lower limits of each weight and bias to minimize the width of the predicted interval while capturing a predefined amount of data within each interval. Khan and Valeo [19] refined this method by utilising the relationship between possibility theory and probability theory, known as probability-possibility transformations (see [48][49][50][51][52]) for details). Khan and Valeo [19] adopted the transformation proposed by Dubois et al [53], where the possibility is viewed as the upper envelope of the family of probability measures [50,[54][55][56].…”

Bridging gaps between several forms of granular computing

“…See Dubois and Prade (2015) for a more complete introduction to the use of the four set functions in possibility theory.…”