On the Properties of the Intersection Probability

In this paper, drawing inspiration from the commutativity results which hold for a number of Bayesian approximations of belief functions (like pignistic function and relative plausibility of singletons) we study the properties of a new probabilistic approximation of belief functions derived from geometric methods: the intersection probability. The intersection probability inherits its name from the fact that, when combined with a Bayesian function through Dempster’s rule, it is equivalent to the intersection of the line joining a pair of belief and plausibility functions with the affine space of Bayesian pseudo belief functions. Its relation with the convex closure operator in the Cartesian space is analyzed, and equivalent conditions under which they commute are given, showing its similarity with orthogonal projection and pignistic transformation. A thorough analysis of the distance between intersection probability and pignistic function in a case study is conducted, and stringent equivalence relations in terms of mass equi-distribution inferred from it.