[Paper Review] Generalizing the Dempster-Shafer Theory to Fuzzy Sets

John Yen|arXiv (Cornell University)|Mar 27, 2013

Multi-Criteria Decision Making3 citations

TL;DR

This paper generalizes the Dempster-Shafer theory to handle fuzzy sets by formulating belief and plausibility functions through probability minimization under constraints derived from basic probability assignments to fuzzy focal elements. By decomposing fuzzy sets into consonant non-fuzzy elements and leveraging possibility theory, the approach preserves the probabilistic interpretation of belief and plausibility while enabling the integration of fuzzy and probabilistic uncertainty in AI systems.

ABSTRACT

With the desire to apply the Dempster-Shafer theory to complex real world problems where the evidential strength is often imprecise and vague, several attempts have been made to generalize the theory. However, the important concept in the D-S theory that the belief and plausibility functions are lower and upper probabilities is no longer preserved in these generalizations. In this paper, we describe a generalized theory of evidence where the degree of belief in a fuzzy set is obtained by minimizing the probability of the fuzzy set under the constraints imposed by a basic probability assignment. To formulate the probabilistic constraint of a fuzzy focal element, we decompose it into a set of consonant non-fuzzy focal elements. By generalizing the compatibility relation to a possibility theory, we are able to justify our generalization to Dempster's rule based on possibility distribution. Our generalization not only extends the application of the D-S theory but also illustrates a way that probability theory and fuzzy set theory can be combined to deal with different kinds of uncertain information in AI systems.

Motivation & Objective

• To extend the Dempster-Shafer theory to handle imprecise and vague evidence in real-world problems by incorporating fuzzy sets.
• To preserve the interpretation of belief and plausibility as lower and upper probabilities in the generalized framework.
• To develop a method for computing belief and plausibility for fuzzy sets using probabilistic constraints derived from basic probability assignments.
• To unify probability theory and fuzzy set theory for handling mixed types of uncertainty in artificial intelligence systems.
• To justify the generalization of Dempster's rule using possibility distributions and compatibility relations.

Proposed method

• Fuzzy focal elements are decomposed into a set of consonant non-fuzzy focal elements to define probabilistic constraints.
• The degree of belief in a fuzzy set is computed as the minimum probability over all consistent probability measures satisfying the constraints.
• The compatibility relation between fuzzy sets and probability measures is generalized using possibility theory to model uncertainty.
• Possibility distributions are used to represent the plausibility of fuzzy sets, ensuring consistency with the generalized belief function.
• Dempster's rule is extended by applying it to the consonant focal elements derived from fuzzy sets, preserving the structure of the original theory.
• The framework ensures that belief and plausibility retain their interpretation as lower and upper probabilities in the fuzzy context.

Experimental results

Research questions

• RQ1 How can the Dempster-Shafer theory be extended to handle fuzzy sets while preserving the interpretation of belief and plausibility as lower and upper probabilities?
• RQ2 What is the appropriate way to define a basic probability assignment for fuzzy focal elements in a generalized evidence theory?
• RQ3 How can the compatibility between fuzzy sets and probability measures be formalized to support uncertainty propagation?
• RQ4 In what way can possibility theory be used to justify the generalization of Dempster's rule to fuzzy sets?
• RQ5 How can the integration of probability and fuzzy set theories enable more robust uncertainty modeling in AI systems?

Key findings

• The proposed method successfully generalizes the Dempster-Shafer theory to fuzzy sets by defining belief and plausibility as minimum and maximum probabilities under constraints.
• The decomposition of fuzzy focal elements into consonant non-fuzzy elements enables the application of standard D-S theory components to fuzzy contexts.
• The use of possibility distributions allows for a principled justification of the generalized Dempster's rule in the fuzzy setting.
• The framework maintains the key interpretation of belief and plausibility as lower and upper probabilities, even when dealing with fuzzy evidence.
• The approach provides a coherent mechanism for combining fuzzy and probabilistic uncertainty, enhancing applicability in real-world AI systems.
• The generalization is validated through theoretical consistency and alignment with existing principles of evidence theory and possibility theory.