  The goal of this paper is to integrate the notions of stochastic conditional
independence and variation conditional independence under a more general notion
of extended conditional independence. We show that under appropriate
assumptions the calculus that applies for the two cases separately (axioms of a
separoid) still applies for the extended case. These results provide a rigorous
basis for a wide range of statistical concepts, including ancillarity and
sufficiency, and, in particular, the Decision Theoretic framework for
statistical causality, which uses the language and calculus of conditional
independence in order to express causal properties and make causal inferences.
