Representing Rules as Random Sets, I: Statistical Correlations between Rules

Abstract I. R. Goodman, H. T. Nguyen, and others have proposed the theory of random sets as a unifying paradigm for evidence theory. The more types of ambiguous evidence that can be brought under the random-set umbrella, the more potentially useful the theory becomes as a systematic methodology for comparing and fusing seemingly incongruent kinds of information. One such type of evidence, conditional events, provides a general framework for dealing with rule-based evidence in a manner consistent with probability theory. This and a companion paper bring conditional event logic—and through it, rules and iterated rules—under the random-set umbrella. We show that Goodman-Nguyen-Walker (GNW) conditional events can be represented as random sets in such a way that the AND, OR, and NOT connectives correspond to intersection, union, and complementation of random sets. We show that GNW logic is the only conditional event algebra that can be “simply” embedded in random sets in this manner. We show that GNW logic generalizes to an entire family of “partial” such logics, each one of which corresponds to a different assumption about the statistical correlations between rules.