Conditional event algebra: a new approach

Recently, algebras of conditional expressions or conditional events compatible with all possible conditional probability evaluations have begun to be developed. Following a brief overview of the main results, two new properties are also demonstration extension of the standard normal disjunctive form expansion to a conditional event setting and a basic connection between the two leading conditional event algebras, showing that one can be considered a weighted averaging of the other in an algebraic sense. However, because of the inherent non-Boolean structure of all of the current approaches, a number of difficulties have arisen, including the issues of higher order conditioning modeling, defining of conditional random variables, and relations to likelihood forms via standard numerically-oriented probability. By taking a fresh approach, based on a product probability space formation, a sounder theory can be established, though it entails significantly more computations.>