On the Axiomization of Conditional Independence

Abstract

Presents an investigation of some aspects of the axiomization of conditional independence of probability. Contributes to the understanding of Pearl's completeness conjecture and identifies a direction for revision which could remove some of the difficulties of Pearl's axiom set, but this alternative is not without its own difficulties. This approach largely simplifies the definition of conditional independence. Then shows the completeness conjecture to be incorrect by presenting counter examples; a new axiom based on the counter examples follows. Discusses the reason for the conjecture being incorrect. Notes that an alternative conjecture could be suggested, but this raises many new questions and increases complexity. Finally, shows that “the disjointness condition on variable sets” is in fact necessary. Concludes that while the axiomization of conditional independence has attractions, it is probably too complex to be pragmatic. Suggests an alternative way forward in inexact reasoning.