Research Project: Decision making with belief functions
Credal semantics of Bayesian transforms
Game-theoretical semantics of epistemic transforms
Decision making with generalized k-additive pignistic transforms
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. Relative belief of singletons, relative plausibility of singletons, and intersection probability can all be described as the foci of different pairs of simplices in the region of all probability measures.
Probability transforms of the epistemic family 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 consider 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 start by conjecturing the shape of the polytope of all the k-additive belief functions dominating a given belief function, to compute the analytical form of the barycenter of the polytope of k-additive dominating belief functions.
Relevant papers: