Statistical Identification of Primary Adjustment Sets From Data

Let $V$ be a finite set of observed variables, and let $G=(V,E_{\to},E_{\leftrightarrow})$ be an acyclic directed mixed graph (ADMG): $E_{\to}$ contains directed edges $A\to B$, $E_{\leftrightarrow}$ contains bidirected edges $A\leftrightarrow B$, and there is no directed cycle. For $A\in V$, let $\mathrm{de}_G(A)$ be its descendants and $\mathrm{an}_G(A)$ its ancestors; extend to sets by unions.

A path is called a confounding arc between distinct $A,B\in V$ if it has no collider and has an arrowhead at both endpoints. Given $C\subseteq V\setminus\{A,B\}$, such a confounding arc is unblocked given $C$ if none of its non-endpoint vertices belongs to $C$. Write $A \not\!\perp\!\!\!\perp_{\mathrm{arc}} B\mid C$ if there exists an unblocked confounding arc between $A$ and $B$ given $C$, and write $A \perp\!\!\!\perp_{\mathrm{arc}} B\mid C$ otherwise.

For distinct $A,B\in V$, a set $C\subseteq V\setminus\{A,B\}$ is an adjustment set for $(A,B)$ if $C\cap(\mathrm{de}_G(A)\cup \mathrm{de}_G(B))=\varnothing$. Given a baseline set $S\subseteq V\setminus\{A,B\}$, an adjustment set $C$ is primary for $(A,B)$ relative to $S$ if

It is minimal primary if no proper subset of $C$ is primary relative to $S$.

Define

fix a deterministic selector $\mathfrak t$ on nonempty finite sets, and define

where $\bot$ is a convention meaning "no minimal primary adjustment set exists." For the target pair $(X,Y)$, let $P^\star(X,Y;G):=P^\star_G(X,Y;\varnothing)$.

Data are generated from an unknown causal model whose latent projection on $V$ is $G$. Let $\mathbb P$ denote either (i) the observational distribution on $V$, or (ii) a specified family of interventional distributions $\{\mathbb P^{do(I=i)}: i\in\mathcal I\}$ for a known intervention set $\mathcal I$. Let $\mathcal G(\mathbb P)$ be the class of ADMGs compatible with the available distribution(s).