Generalizing the Dempster-Shafer Theory to Fuzzy Sets (1304.2383v1)

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.

• The paper introduces a novel generalization of Dempster-Shafer theory by incorporating fuzzy set and possibility principles.
• The paper develops an optimization-based belief function and decomposes fuzzy focal elements to compute belief and plausibility measures efficiently.
• The paper generalizes Dempster’s combination rule, enabling robust handling of conflicting and vague evidence in AI applications.

Generalizing the Dempster-Shafer Theory to Fuzzy Sets

The research paper by John Yen presents a sophisticated extension of the Dempster-Shafer (D-S) theory to accommodate fuzzy sets, aiming to address the limitations in handling vague concepts and imprecise evidence that are commonplace in complex real-world applications. This paper proposes a rigorous enhancement to the conventional D-S theory by integrating the principles of fuzzy set theory and possibility theory, preserving the essential characteristics of belief and plausibility as lower and upper probabilities.

Key Contributions

1. Generalized Compatibility Relations: A fundamental component of the D-S theory is the compatibility relation, which the author extends to a fuzzy relation representing joint possibility distributions. This generalization allows for expressions of degrees of possibility between elements across spaces, which is more aligned with real-world uncertainty where interactions are often a matter of degree rather than binary outcomes.
2. Optimization-Based Belief Function: By viewing belief functions as outcomes of an optimization problem constrained by a basic probability assignment (BPA), the paper derives belief functions for fuzzy sets via a modified objective function. The decomposition of fuzzy focal elements into consonant non-fuzzy focal elements plays a critical role in this formulation.
3. Decomposition and Combination of Fuzzy Focal Elements: Yen's approach involves decomposing fuzzy focal elements into non-fuzzy components, simplifying the application of probability constraints and facilitating the computation of belief and plausibility measures. This decomposition enables the generalization of Dempster's combination rule to incorporate fuzzy sets, allowing the combination of uncertain information from multiple sources even when the evidence is partially conflicting.
4. Norm of Subnormal Fuzzy Focal Elements: The paper addresses the normalization of subnormal fuzzy focal elements that emerge from the combination of fuzzy BPAs, ensuring that the probabilistic framework remains intact and prevents allocation of probability mass to empty sets.

Implications and Future Directions

The integration of fuzzy logic into the D-S framework provides a robust foundation for expert systems and AI applications dealing with vague and imprecise evidence. By maintaining the structure of belief and plausibility functions as lower and upper probability bounds, the author successfully preserves the probabilistic semantics of the D-S theory while extending its applicability to a broader range of uncertainties encountered in AI systems.

This research also strengthens the interaction between probability theory and fuzzy set theory, suggesting a hybrid reasoning approach under uncertainty that could be explored further. The methodology could lead to advances in fields requiring nuanced uncertainty assessment, such as risk analysis, sensor fusion in robotics, and decision-making processes in AI.

Future research may focus on enhancing computational efficiency and exploring different domain applications. Additionally, refining the generalization of inclusion operators and investigating the impact of different fuzzy operators on belief functions could expand the theoretical foundation established in this paper.

In conclusion, John Yen's work provides substantial advancements in evidential reasoning under uncertainty, establishing a rigorous mechanism to address the challenges of imprecise and vague information in complex systems. This paper lays down a crucial pathway for AI researchers interested in merging discrete probabilistic reasoning and the continuous nature of fuzzy logic.