Approximate inference and interval probabilities

Abstract

This work presents results of research to develop of approximate inference formulas in the context of the calculus of evidence of Dempster-Shafer and the theory of interval probabilities. These formulas generalize the probabilistic domain the well-known relations between the truth values of the antecedent and the consequent propositions of a classical implication P → Q.

Approximate conditional knowledge is assumed to be expressed either as sets of possible values (actually numeric intervals) of conditional probabilities, or as the Shafer belief functions induced by certain restricted multivalued mappings — the nature of the restriction representing the truth of the antecedent in an implication — between a space representing possible states of the real world and another space used to represent propositional truth.

A notion of consistence between unconditional and conditional probability intervals is introduced to represent agreement of the constraints induced by both types of distributions. The integration of conditional and unconditional knowledge is accordingly described as the refinement of probability interval estimates for unconditional propositions so as to achieve consistence with conditional constraints. This refinement always results in interval-valued distributions which are more specific than the corresponding unmodified estimates.

Formulas for conditional knowledge integration are discussed, together with the computational characteristics of the methods derived from them. Of particular importance is one such evidence-integration formalism produced under a belief function interpretation — generalizing both modus ponens and modus tollens inferential mechanisms — which integrates conditional and unconditional knowledge without resorting to iterative or sequential approximations. Further, this formalism produces Shafer mass distributions as output using similar distributions as input.