On Transformations between Belief Spaces

Abstract: Commutative monoids of belief states have been defined by imposing one or more of the usual axioms and employing a combination rule. Familiar operations such as normalization and the Voorbraak map are surjective homomorphisms. The latter, in particular, takes values in a space of Bayesian states. The pignistic map is not a homomorphism between these same spaces. We demonstrate an impact this may have on robust decision making for frames of cardinality at least 3. We adapt the measure zero reflection property of some maps between probability spaces to define a category of belief states having plausibility zero reflecting functions as morphisms. Our definition encapsulates a generalization of the notion of absolute continuity to the context of belief spaces. We show that the Voorbraak map is a functor valued in this category.