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Geometry of relative plausibility and belief of singletons

The study of the interplay between belief and probability has recently been posed in a geometric framework, in which belief and plausibility functions are represented as points of simplices in a Cartesian space. All Bayesian approximations of a belief function b form two homogeneous groups, which we call “affine” and “epistemic” families. In this paper, in particular, we focus on relative plausibility and belief of singletons and show that they form, together with a new Bayesian function called “non-Bayesianity flag”, a homogenous family of Bayesian functions related to b, in terms of both their geometry and their behavior with respect to Dempster’s rule of combination. We investigate their geometry, which turns out to be described in terms of three planes and angles. We combine algebraic and geometric properties of the relative plausibility function to conjecture its new interpretation as solution of the probabilistic approximation problem formulated in terms of the rule of combination. Finally, drawing inspiration from the binary case, we prove that all Bayesian approximations of both families coincide when belief functions assign the same mass to events of the same size.