Two k-additive generalizations of the pignistic transform for imprecise decision making

The Transferable Belief approach to the Theory of Evidence is based on the pignistic transform which, mapping belief functions to probability distributions, allows to make “precise” decisions on a set of disjoint hypotheses via classical utility theory. In certain scenarios, however, such as medical diagnosis, the need for an “imprecise” approach to decision making arises, in which sets of possible outcomes are compared. We propose here a framework for imprecise decision derived from the TBM, in which belief functions are mapped to k-additive belief functions (i.e., belief functions whose focal elements have maximal cardinality equal to k) rather than Bayesian ones. We do so by introducing two alternative generalizations of the pignistic transform to the case of k-additive belief functions. The latter has several interesting properties: depending on which properties are deemed the most important, the two distinct generalizations arise. The proposed generalized transforms are empirically validated by applying them to imprecise decision in concrete pattern recognition problems.