Transitional Conditional Independence

Abstract

Statistical models contain variables that are not random: parameters, treatments, environments, design points. Ordinary conditional independence cannot express relations involving such variables. To apply it one must first put a distribution on them, and that changes the meaning of the statement. This paper introduces transitional conditional independence. It relates three variables on a Markov kernel $K(W|T)$ with non-stochastic input $T$, and is defined by a single factorization: \[ X\perp\!\!\perp_{K(W|T)} Y |Z \quad :\iff \quad \exists\, Q(X|Z):\; K(X,Y,Z|T) = Q(X|Z)\otimes K(Y,Z|T).\] The relation asserts a Markov kernel $Q(X|Z)$ that is the same for every input $t$. It therefore yields a factorization rather than an almost-sure identity between conditional expectations, and it needs no distribution on the input space. The relation is asymmetric. We show that the asymmetry is essential: symmetrizing it destroys the statements it was built to make. We prove left and right versions of all separoid rules except Symmetry. Ten of them hold on arbitrary measurable spaces, the remaining ones under one condition on the spaces involved, and we give criteria for when Symmetry itself holds. We axiomatize the resulting structure and show that it arises from any symmetric separoid by a shift. We give several applications. Ancillarity, sufficiency and adequacy become factorizations that hold pointwise in the parameter, without a prior and without null sets; the theorems of Fisher--Neyman and of Basu take this form. The invariance hypothesis of invariant prediction, $Y \perp\!\!\perp E | X_S$, receives its intended meaning: one kernel predicts $Y$ from $X_S$ in every environment $E$. And Bayesian networks with non-stochastic input nodes satisfy a directed global Markov property whose graphical id-separation criterion returns a factorization of Markov kernels, on arbitrary input spaces.