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Two new Bayesian approximations of belief functions based on convex geometry
1Perception Project, INRIA Rhône-Alpes, 38334 Saint Ismier, France. Fabio.Cuzzolin@inrialpes.fr
Summary
This study geometrically analyzes belief functions and probability functions within the theory of evidence. It identifies key geometric entities and their associated probability measures, offering new insights into belief representation.
Area of Science:
- Decision Theory
- Probability Theory
- Geometric Analysis
Background:
- The theory of evidence deals with uncertainty and belief representation.
- Existing approaches often lack a clear geometric interpretation.
- Understanding the relationship between belief and probability functions is crucial.
Purpose of the Study:
- To geometrically analyze the relationship between belief functions and probability functions.
- To identify and study key geometric entities in the theory of evidence.
- To provide interpretations of these geometric entities in terms of degrees of belief.
Main Methods:
- Geometric approach to the theory of evidence.
- Analysis of binary domains.
- Identification of geometric entities: dual line, orthogonal complement, simplex.
- Study of orthogonal projection and intersection probability.
Main Results:
- Identified three major geometric entities relating belief functions (b.f.) to probability sets (P).
- Associated each geometric entity with a unique probability measure derived from the b.f.
- Characterized the geometry and properties of the orthogonal projection of a b.f. onto P.
Conclusions:
- The geometric approach offers a novel perspective on belief and probability functions.
- Geometric entities provide a framework for interpreting degrees of belief.
- Further research can explore affine combinations and their implications.
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