A credal representation of the interval probability associated with a belief function can be proposed. It relates to several classical
Bayesian transformations of BFs through the notion of 'focus' of a pair of simplices. While a belief function corresponds to a polytope
of probabilities consistent with it, the related interval probability is geometrically represented by a pair of upper and lower simplices.
Starting from the interpretation of the pignistic function as the center of mass of the credal set of consistent probabilities, we prove that
the relative belief of singletons, the relative plausibility of singletons, and the intersection probability can all be described as the foci
of different pairs of simplices in the region of all probability measures. The formulation of frameworks similar to the transferable belief model
for such Bayesian transformations appears then at hand.
Probability transformations of the epistemic family possess natural rationales within Shafer's formulation of the theory of evidence, while they
are not consistent with the credal or probability-bound semantic of belief functions. We can prove, however, that these transforms can be given
in this latter case an interesting rationale in terms of optimal strategies in a
non-cooperative game. In particular, they can be seen as tools
to provide optimal conservative strategies in a maximin/minimax 2-person game scenario
derived from Wald's model, in which a player has to optimize
their minimal expected gain/maximal expected loss under epistemic uncertainty in the form of a belief function.
Inspired by the classical results by Jaffray on the credal set of probability measures dominating any given belief function,
we considered the dominance properties of the set of the pignistic k-additive belief functions, i.e., BFs with focal elements of
cardinality up to k.
We started by conjecturing the shape of the polytope of all the k-additive belief functions dominating a given belief
Under such conjecture, we computed the analytical form of the barycenter of the polytope of k-additive dominating belief functions,
and we studied the location of the pignistic k-additive belief functions with respect to this polytope and its barycenter.