On transformations between belief spaces

Abstract

Commutative monoids of belief states have been defined by imposing one or more of the usual axioms and assigning a combination rule. Familiar operations such as normalization and the Voorbraak map are surjective homomorphisms. The latter map takes values in a monoid of Bayesian states. The pignistic map is not a monoid homomorphism. This can impact robust decision making for frames of cardinality at least 3. We adapt the concept of measure zero reflecting functions between probability spaces to define a category P0R having belief states as objects and plausibility zero reflecting functions as morphisms. This definition encapsulates a generalization of the notion of absolute continuity to the context of belief spaces. We show that the Voorbraak map induces a functor valued in P0R that is right adjoint to the embedding of Bayesian states. © 2008 Springer-Verlag Berlin Heidelberg.