Evidential Reasoning in a Categorial Perspective: Conjunction and Disjunction of Belief Functions

The categorial approach to evidential reasoning can be seen as a combination of the probability kinematics approach of Richard Jeffrey (1965) and the maximum (cross-) entropy inference approach of E. T. Jaynes (1957). As a consequence of that viewpoint, it is well known that category theory provides natural definitions for logical connectives. In particular, disjunction and conjunction are modelled by general categorial constructions known as products and coproducts. In this paper, I focus mainly on Dempster-Shafer theory of belief functions for which I introduce a category I call Dempster's category. I prove the existence of and give explicit formulas for conjunction and disjunction in the subcategory of separable belief functions. In Dempster's category, the new defined conjunction can be seen as the most cautious con junction of beliefs, and thus no assumption about distinctness (of the sources) of beliefs is needed as opposed to Dempster's rule of combination, which calls for distinctness (of the sources) of beliefs.