Toward a comprehensive theory of linguistic and probabilistic evidence: two new approaches to conditional event algebra

This paper first surveys the basic state-of-the art in modeling and combining conditional information, compatible with natural conditional probability evaluations via conditional event algebras. A critical analysis of the situation is given, where current approaches are based upon interpreting conditional events as event intervals. This includes the present problems of modeling conditional random variables, higher order conditionals, and the general incompatibility of these conditional event algebras with the traditional numerically-oriented approach using conditional likelihoods for the independent source case. To address these issues, two new approaches to conditioning are presented here. The first is based upon extending the usual arithmetic division operation to the case of zero denominators and is shown to lead to positive results for the above-mentioned higher order conditioning problem, as well as to a natural way to define fuzzy conditional sets. The second new approach is based upon a joint countable product space representation and not only leads to solutions to all three above problems, but also leads to reasonable definitions of fuzzy conditional sets. One drawback of this approach, however, is the increased computational complexity entailed in its implementation. This issue is also addressed in part. Finally, the techniques are shown to yield a more comprehensive way of dealing with either, or both, linguistic-based evidence and stochastic evidence