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.