A mathematical formulation of uncertain information

Abstract

This paper introduces a mathematical model of uncertain information. Each body of uncertain information is an information quadruplet, consisting of a code space, a message space, an interpretation function, and an evidence space. Each information quadruplet contains prior information as well as possible new evidence which may appear later. The definitions of basic probability and belief function are based on the prior information. Given new evidence, Bayes' rule is used to update the prior information. This paper also introduces an idea of independent information and its combination. A combination formula is derived for combining independent information. Both the conventional Bayesian approach and Dempster-Shafer's approach belong to this mathematical model. A Bayesian prior probability measure is the prior information of a special information quadruplet; Bayesian conditioning is the combination of special independent information. A Dempster's belief function is the belief function of a different information quadruplet; the Dempster combination rule is the combination rule of independent quadruplets. This paper is a mathematical study of handling uncertainty and shows that both the conventional Bayesian approach and Dempster-Shafer's approach originate from the same mathematical theory.