I. R. Goodman

It is an understatement that both the theory and applications of probability conditional or unconditional play an essential role in the processing and use of disparate information in decision-making in C4I systems. Apropos to the theme of this symposium, Making Information Superiority Happen, the paper outlined here describes new applications, insights, and theoretical aspects of ongoing work by the authors toward improving the rationale for use of probability theory, keeping in mind issues of scalability and computational complexity. This paper extends the ideas first presented in last year's CCRTS at Newport, RI. In short, the mathematical theme of this paper is both a summary of past research efforts together with new results on the problem of best estimating partially specified conditional and unconditional probabilities of interest via a second order bayesian probability approach. Among the new derivations provided in this paper is a significant reduction in computational effort in obtaining (again, in the second order probability sense) optimal or near-optimal probability estimates, all within the setting of a boolean conditional event algebra which allows full compatibility with conditional probability evaluations.

Beginning with work in the mid 1970's and early 1980's, it was discovered that fundamental homomorphic-like relations exist between many first order fuzzy logic concepts and naturally corresponding probability ones via the one-point coverage events for appropriately chosen random subsets of the domains of the fuzzy sets considered. This paper first extends and modifies the above-mentioned homomorphic-like relations previously established. It also introduces a number of new homomorphic-like relations between fuzzy logic concepts and probability, utilizing two recently derived subfields of probability theory: conditional and relational event algebra. In addition, a newly invigorated branch of probability theory dealing with second order probabilities (or `probabilities of probabilities') is shown to be applicable to treating certain deduction problems involving conditioning of populations.

This paper concludes the work begun in Part 1 in presenting a coherent theory of conditioning consistent with all conditional probability evaluations.