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An Interpretation of Conflicting Parts of Belief Functions on Two-Element Frame of Discrement
Daniel, Milan
When combining belief functions by the conjunctive rules of combination, conflicts often appear, which are assigned to empty set by un-normalized conjunctive rule or normalized by Dempster's rule of combination. This contribution is devoted to an interpretation of the conflicting part of a belief function on a two-element frame of discernment. It is based on the author's idea of the unique decomposition of such function into its conflicting and non-conflicting part (CJS 2010, Otaru). A relation of conflicting part of a belief function to internal conflict of the function is also studied and a new definition of internal conflict is introduced. New internal conflict is compared with the previous approaches. Keywords: Belief function, Dempster-Shafer theory, uncertainty, Dempster's semigroup, internal conflict, conflict between belief functions, non-conflicting part of belief function, conflicting part of belief function.
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Introduction to Algebra of Belief Functions on Three-element Frame of Discernment - A General Case
Daniel, Milan
This contribution presents the second part of the introductive study of algebraic structure of belief functions (BFs) on 3-element frame of discernment. Algebraic method by Hájek & Valdés for BFs on 2-element frames is generalized to larger frame of discernment. Due to complexity of the algebraic structure, the study is divided into 2 parts, the present one is devoted to a case of general BFs. The definition of Dempster's semigroup (an algebraic structure) of BFs on 3-element frame is recalled from the first part of the study. Results related to Bayesian and quasi Bayesian BFs from the first part are also briefly recalled. Further substructures related to another subsets of general BFs are described and analyzed (including idempotents, simple complementary BFs, generalizations of subsemigroups of simple BFs) and subalgebras isomorphic to Dempster's semigroup on 2-element frame of discernment. Ideas and open problems for future research are presented.
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Some Results on Set-Valued Possibilistic Distributions
Kramosil, Ivan
When proposing and processing uncertainty decision making algorithms of various kinds and purposes we meet more and more often probability distributions ascribing to random events non-numerical uncertainty degrees. The reason is that we have to process systems of uncertainties for which the classical conditions like sigma-additivity or linear ordering of values are too restrictive to define sufficiently closely the nature of uncertainty we would like to specify and process. For the case of non-numerical uncertainty degrees at least the two criteria may be considered. First systems with rather complicated, but sophisticated and nontrivially formally analyzable uncertainty degrees. E.g., uncertainties supported by some algebras or partially ordered structures. Contrary, we may consider more easy non-numerical, but on the intuitive level interpretable relations. Well-known examples of such structures are set-valued possibilistic measures. Some perhaps interesting particular results in this direction will be introduced and analyzed in the contribution.
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Morphisms of Dempster's Semigroup: A Revision and Interpretation
Daniel, Milan
Analyzing three new approaches to interpretation, definition, and measurement of conflicts of belief Functions (BFs), we previously observed a possibility of expression of a BF Bel as Dempster's sum of non-conflicting BF Bel0 with the same plausibility decisional support as the original BF Bel has and of indecisive BF BelS which does not prefer any of the elements of frame of discernment. Based on this observation, the theory of homomorphisms of Dempster's semigroup (the algebra of non-extremal BFs on 2-element frame of discernment with Dempster's rule as its binary operation) is updated in the present contribution. New homomorphisms are introduced; their interpretation and relation to the original ones are presented.
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