Alternative approaches to the widely known pignistic transformation of belief functions are presented and analyzed. Pignistic, cautious, proportional, and disjunctive probabilistic transformations are examined from the point of view of their interpretation, of decision making and from the point of view! of their commutation with rules~operators! for belief function combination. A relation to the plausibility probabilistic transformation is added.

The combination ofpossibly conpicting beliefs andevidence forms an imporrantpan of various disciplines of arrijicial reasoning. In everyday discourse dogmatic be- liefs are expressed by observers when they have a strong and rigid opinion abour a subject of interest. Such beliefs can be expressed and formalised within the Demspter-Shafer belief theory. This paper describes and compares methods for combining dogmatic or highly conpicting beliefs within this framework.classic assumption that elements in a frame of discernment must be mutually exclusive, and thereby provides a new interpretation and framework for managing belief conflicts.In this paper, we describe various proposals for combining beliefs and how they handle cases when the beliefs are conflicting and dogmatic. Section 2 presents a background on Dempster-Shafer (DS) theory of evidence and Section 3 describes Dempster's rule of combination as well as its unnonnalised and disjunctive versions. The subsequent sections introduce alternatives lo Dempster's rule i.e. the Weighted Operator (Section 4), the minC Combination (Section 5). and the Consensus Operator (Section 6). In

The nature of a contradiction (conflict) between two belief functions is investigated. Alternative ways of distributing the contradiction among nonempty subsets of frame of discernment are studied. The paper employes a new approach to understanding contradictions and introduces an original notion of potential contradiction. A method of an associative combination of generalized belief functions -minC combination and its derivation -is presented as part of the new approach. A proportionalization of generalized results is suggested as well. IntroductionWhen applying Dempster rule of combination to two belief functions, one part of the belief is assigned to nonempty subsets of the frame of discernment X and the other part is assigned to the empty set. This second part represents a contradiction (a conflict) between the sources of the two belief functions. A certain amount of the contradiction (conflict) usually arises even when combining the same belief function with itself.To obtain a sum of belief masses assigned to the nonempty subsets of X equal to one, i.e., to obtain a belief function as defined by Shafer [12], the result of the combination is usually normalized. Such a normalization is not generally required. Thus, for example in the TBM, the unnormalized Dempster rule of combination is used [13], [14].The problem with normalization lies within finding a way to reallocate the part of the belief that represents the contradiction (conflicts) among the nonempty subsets of the frame of discernment.Another approach to avoiding normalization is to assign the whole contradictive belief mass to the whole frame of discernment; see e.g. [17].The need for an associative combination rule for belief functions is described in Sect. 2. It is followed by the presentation of the used abbreviations, preliminaries, introductory examples and a description of the present state of art. The process of looking for an associative combination of generalized belief functions is presented in Sect. 3. Section 4 introduces different types of contradiction, the notion of a potential contradiction, and a new idea of how to consider a contradiction as well. An associative operation of combination applicable to generalized belief functions on a three-element frame of discernment is constructed in Sect. 5. Lemma on how to combine different types of contradictions are presented in the same section. Section 6 presents ideas for future work. A need for an associative combination of belief functionsIn belief function theory, there are many different operations that can be applied to beliefs. In this paper, we consider neither conditioning nor actualization of old beliefs by new ones with priority given to either old or to new.In this paper we are interested in situations in which there are several beliefs, represented by belief functions, without priority given to any of them. In such a situation, the combination rule must be associative and commutative. Dempster rule of combination satisfies these requirements, but examples of its weaknesses are p...

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