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About Two Consonant Conflicts of Belief Functions
Daniel, M. ; Kratochvíl, Václav
General belief functions usually bear some internal conflict which comes mainly from disjoint focal elements. Analogously, there is often some conflict between two (or more) belief functions. After the recent observation of hidden conflicts (seminar CJS’17 [17]), appearing at belief functions with disjoint focal elements, importance of interest in conflict of belief functions has increased. This theoretical contribution introduces a new approach to conflicts (of belief functions). Conflicts are considered independently of any combination rule and of any distance measure. Consonant conflicts are based on consonant approximations of belief functions in general; two special cases of the consonant approach based on consonant inverse pignistic and consonant inverse plausibility transforms are discussed. Basic properties of the newly defined conflicts are presented, analyzed and briefly compared with our original approaches to conflict (combinational conflict, plausibility conflict and comparative conflict), with the recent conflict based on non-conflicting parts, as well as with W. Liu’s degree of conflict.
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Indecisive Belief Functions
Daniel, Milan
This study presents an idea of indecisive functions, their general and also special definitions, plausibility and pignistic indecisive belief functions. The rich structure of indecisive belief functions is studied in general, and also in special views: both general substructures and indecisive belief functions on three-element and general finite frames of discernment. We are focused to pignistic and contour (plausibility) indecisive belief functions, including their mutual relationship in our study. The later have interesting algebraic structure related to Dempster’s rule of combination.
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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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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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A Note on Factorization of Belief Functions
Jiroušek, Radim ; Shenoy, P. P.
The paper compares two main types of factorization of belief functions (one based on the Dempster´s rule of combination, the other based on the operator of composition) and shows that both the approaches are equivalent to each other in case of unconditional factorization and shows what are the differences when overlapping factorization is studied.
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Podmiňování domnění v DSmT
Daniel, Milan
Stručné připomenutí základních pojmů z Dempster-Shaferovy teorie i DSmT (Dezert-Smarandacheovy teorie) je následováno připomenutím stávajících výsledků o DSm podmiňování. DSm pravidla pro podmiňování domnění jsou analyzována analogicky analýze podmiňování domnění v klasickém případě. V příspěvku je definováno nové zobecněné "plain" podmiňovací pravidlo a jsou zde prezentovány zobecněné formule pro DSm pravidla pro podmiňování domnění. Dále je zavedeno negativní podmiňování v DSm svazové struktuře. Je zde definováno negativní "plain" a negativní Dempsterovo podmiňovací pravidlo a jsou prezentovány první výsledky o těchto nových pravidlech. Závěrem je nastíněna široká oblast otevřených problémů negativního DSm podmiňování a jeho vztah k pozitivnímu podmiňování.
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