Foundations of probabilistic inference with uncertain evidence

Abstract

The application of formal inference procedures, such as Bayes Theorem, requires that a judgment is made, by which the evidential meaning of physical observations is stated within the context of a formal model. Uncertain evidence is defined as the class of observations for which this statement cannot take place in certain terms. It is a significant class of evidence, since it cannot be treated using Bayes Theorem in its conventional form [G. Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, NJ, 1976]. In this paper, we present an extension of the Bayesian theory that can be used to perform probabilistic inference with uncertain evidence. The extension is based on an idealized view of inference in which observations are used to rule out possible valuations of the variables in a modeling space. The extension is different from earlier probabilistic approaches such as Jeffrey’s rule of probability kinematics and Cheeseman’s rule of distributed meaning, by introducing two forms of evidential meaning representation are presented, for which non-probabilistic analogues are found in theories such as Evidence Theory and Possibility Theory. By viewing the statement of evidential meaning as a separate step in the inference process, a clear probabilistic interpretation can be given to these forms of representation, and a generalization of Bayes Theorem can be derived. This generalized rule of inference allows uncertain evidence to be incorporated into probabilistic inference procedures.