On Computing Evidential Centroid Through Conjunctive Combination: An Impossibility Theorem

Abstract

The theory of belief functions (TBF) is now a widespread framework to deal and reason with uncertain and imprecise information, in particular to solve information fusion and clustering problems. Combination functions (rules) and distances are essential tools common to both the clustering and information fusion problems in the context of TBF, which have generated considerable literature. Distances and combination between evidence corpus of TBF are indeed often used within various clustering and classification algorithms, however their interplay and connections have seldom been investigated, which is the topic of this paper. More precisely, we focus on the problem of aggregating evidence corpus to obtain a representative one, and we show through an impossibility theorem that in this case, there is a fundamental contradiction between the use of conjunctive combination rules on the one hand, and the use of distances on the other hand. Rather than adding new methodologies, such results are instrumental in guiding the user among the many methodologies that already exist. To illustrate the interest of our results, we discuss different cases where they are at play. Impact Statement-Within the theory of belief functions, both distances and conjunctive combination rules can be used to achieve very similar purposes: evaluating the conflict between sources, performing supervised or unsupervised learning in presence of evidential information, or more simply obtaining a synthetic representation of multiple items of information. However, the results obtained by both approaches may show some inconsistency between them. This paper provides some insight as to why this may happen, showing that the two approaches are definitely at odds, and that using distances is, for instance, incompatible with some fundamental notions of the theory of belief functions, such as the least commitment principle. We illustrate the importance of the studied differences on problems such as k-centroid clustering, and discuss the importance of interpretations in such problems, which is rarely done in the literature.