α-junctions of categorical mass functions

Abstract

The set of α-junctions is the set of linear associative and commutative combination operators for belief functions. Consequently, the properties of α-junctive rules make them particularly attractive on a theoretic point of view. However, they are rarely used in practice except for the α = 1 case which corresponds to the widely used and well understood conjunctive and disjunctive rules. The lack of success of α- junctions when α < 1 is mainly explained by two reasons. First, they require a greater computation load due to a more complex mathematical definition. Second, the mass function obtained after combination is hard to interpret and sometimes counter-intuitive. Pichon and Denoeux [4] brought a significant contribution to circumvent both of these two limitations. In this article, it is intended to pursue these efforts toward a better understanding of α-junctions. To that end, this study is focused on the behavior of α-junctions when categorical mass functions are used as entries of an α-junctive combination rule. It is shown that there exists a conjunctive and a disjunctive canonical decomposition of the mass function obtained after combination.