A Theory of Structural Independence

Abstract

Structural independence is the (conditional) independence that arises from the structure rather than the precise numerical values of a distribution. We develop this concept and relate it to $d$-separation and structural causal models. Formally, let $U=(U_i{)}_{i\in I}$ be an independent family of random elements on a probability space $(\hat{I}\text{c◯},\mathcal{A},\mathbb{P})$. Let $X$, $Y$, and $Z$ be arbitrary $\ddot{I}ƒ(U)$-measurable random elements. We characterize all independences $X\perp Y\mid Z$ implied by the independence of $U$ and call these independences \textit{structural}. Formally, these are the independences which hold in all probability measures $P$ that render $U$ independent and are absolutely continuous with respect to $\mathbb{P}$; i.e., for all such $P$, it must hold that $X{\perp }_{P}Y\mid Z$. We introduce the history $\mathcal{H}(X\mid Z):\hat{I}\text{c◯}\to \mathcal{P}(I)$, a combinatorial object that measures the dependence of $X$ on $U_i$ for each $i\in I$ given $Z$. The independence of $X$ and $Y$ given $Z$ is implied by the independence of $U$ if and only if $\mathcal{H}(X\mid Z)\cap \mathcal{H}(Y\mid Z)=\varnothing$ almost surely with respect to $\mathbb{P}$. Finally, we apply this $d$-separation-like criterion in structural causal models to discover a causal direction in a toy setting.

39 pages. Updated: Moved well-known theorems, definitions, and infinite product probability measure material to the appendix. Revised main sections for clarity