Transitional Conditional Independence

Abstract

We develope the framework of transitional conditional independence. For this we introduce transition probability spaces and transitional random variables. These constructions will generalize, strengthen and unify previous notions of (conditional) random variables and non-stochastic variables, (extended) stochastic conditional independence and some form of functional conditional independence. Transitional conditional independence is asymmetric in general and it will be shown that it satisfies all desired relevance relations in terms of left and right versions of the separoid rules, except symmetry, on standard, analytic and universal measurable spaces. As a preparation we prove a disintegration theorem for transition probabilities, i.e. the existence and essential uniqueness of (regular) conditional Markov kernels, on those spaces. Transitional conditional independence will be able to express classical statistical concepts like sufficiency, adequacy and ancillarity. As an application, we will then show how transitional conditional independence can be used to prove a directed global Markov property for causal graphical models that allow for non-stochastic input variables in strong generality. This will then also allow us to show the main rules of causal/do-calculus, relating observational and interventional distributions, in such measure theoretic generality.