A THEORY OF CONDITIONAL INFORMATION FOR PROBABILISTIC INFERENCE IN INTELLIGENT SYSTEMS .2. PRODUCT SPACE APPROACH

This paper continues the theme, begun in Part I, for the further development of probability theory to account for the special needs of probabilistic intelligent systems, especially for the modeling and manipulating of conditioned information stemming from disparate sources. In Part I, the interval of events approach to the development of conditional events was reviewed and analyzed. In this paper, a new Cartesian product approach to conditional events is taken. The result is a full Boolean algebra (or sigma-algebra) setting for all results-unlike the essential non-Boolean character of the former approach. In turn, this leads to the solution of a number of outstanding problems in the field, including the representation of higher order conditioning and the development of full conditional random variables. In addition, other applications areas are provided for both the new and previous approaches to conditional events, including Bayesian updating for conditional information, information bounding problems, and combination of inference rules and data. Finally, the more computationally intensive structure of the new approach is compared with the previous approach and some basic approximation procedures are considered for implementations.