A Generalization of Bayes Theorem in Theory of Evidence

Abstract

A generalized Bayes theorem in the frame of Shafer's theory of evidence is proposed. Smets proposed a procedure for obtaining a posterior basic probability assignment from prior and conditional basic probability assignments using Dempster rule of combination. We show that the uncertainty in the sense of non-specificity of a probability distribution changes by the Smets' procedure. Hence, it contradict with Bayes theorem. We formulate a new combination rule and construct a procedure for obtaining a posterior basic probability assignment without changing the uncertainty. We verify that the proposed rule is reduced to Bayes theorem when each of basic probability assignment is a regular probability assignment. It is also Jeffrey's rule when observation is obtained by a probability distribution and prior distributions are probability distributions.