A Theory of Frame Transformations for Belief Combination

Abstract

In approximate reasoning, aggregation of multiple measures representing uncertainty, belief, or desirability may be achieved by defining an appropriate combination operator. Formalisms such as probability theory and Dempster–Shafer evidence theory have proposed specific forms for these operators. Ad-hoc approaches to combination have also been put forth, a classical example being the MYCIN calculus of certainty factors. In the present paper we present an analytical theory of combination operators based on the idea that certain combination operators are characterized by special geometric frames of reference or systems of coordinates in which the operators reduce to the canonical arithmetic sum. The cornerstone of our theory is an algorithm that determines whether a given combination operator can be so reduced, and that explicitly constructs a normalizing reference frame directly from the operator whenever such a frame exists. Our approach provides a natural nonlinear scaling mechanism that extends operators to parameterized families, allowing one to adjust the sensitivity of the operators to new information and to control the asymptotic growth rate of the aggregate values produced by the operators in the presence of an unbounded number of information sources. We also give a procedure to reconstruct the normalizing reference frame directly from the group of nonlinear scaling operations associated with it.