A Theory of Structural Independence
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 $(Ω, \mathcal{A}, \mathbb{P})$. Let $X$, $Y$, and $Z$ be arbitrary $σ(U)$-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 $\mathcal{H}(X \mid Z) : Ω\to \mathcal{P}(I)$, 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 $\mathcal{H}(X \mid Z) \cap \mathcal{H}(Y \mid Z) = \emptyset$ 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.