Front-door Reducibility: Reducing ADMGs to the Standard Front-door Setting via a Graphical Criterion

TL;DR

The paper extends front-door adjustment to a wider class of causal graphs by introducing Front-door Reducibility (FDR), a graphical criterion on ADMGs that aggregates variables into super-nodes and reduces complex settings to a standard front-door form. It proves a graph-level equivalence between the FDR criterion and the existence of an FDR adjustment, and develops FDR-TID, an exact algorithm that identifies admissible FDR triples (X*,Y*,M*). This yields simple, estimable interventional distributions in many graphs where the general ID algorithm would produce unwieldy expressions. By focusing on interpretability and computational tractability while maintaining generality across mixed graphs, FDR complements existing identification methods. The work also discusses non-FDR examples and provides a termination-guaranteed procedure for constructing admissible triples, paving the way for practical deployment and future estimators tailored to FDR graphs.

Abstract

Front-door adjustment gives a simple closed-form identification formula under the classical front-door criterion, but its applicability is often viewed as narrow. By contrast, the general ID algorithm can identify many more causal effects in arbitrary graphs, yet typically outputs algebraically complex expressions that are hard to estimate and interpret. We show that many such graphs can in fact be reduced to a standard front-door setting via front-door reducibility (FDR), a graphical condition on acyclic directed mixed graphs that aggregates variables into super-nodes $(\boldsymbol{X}^{*},\boldsymbol{Y}^{*},\boldsymbol{M}^{*})$. We characterize the FDR criterion, prove it is equivalent (at the graph level) to the existence of an FDR adjustment, and present FDR-TID, an exact algorithm that finds an admissible FDR triple with correctness, completeness, and finite-termination guarantees. Empirical examples show that many graphs far outside the textbook front-door setting are FDR, yielding simple, estimable adjustments where general ID expressions would be cumbersome. FDR therefore complements existing identification methods by prioritizing interpretability and computational simplicity without sacrificing generality across mixed graphs.

Figures (4)

• Figure 1: The example of the equivalence projection from two complicated (but reducible) ADMGs (a) and (b) to a simple, well studied ADMG (c) where ordinary front-door adjustment can be used to obtain the interventional distribution $p\left(\boldsymbol{Y}^{*}\left|do\left(\boldsymbol{X}^{*}\right)\right.\right)$. Different colors show the projection relation of the nodes between (a), (b) and (c), e.g., for (a) $\boldsymbol{X}^{*}=\left\{ X,K\right\} ,$$\boldsymbol{M}^{*}=\left\{ M\right\}$, and for (b) $\boldsymbol{X}^{*}=\left\{ X,U\right\}$, $\boldsymbol{M}^{*}=\left\{ M,Z\right\}$.
• Figure 2: Example ADMGs relative to the cause and effect of interest pair $\left(X,Y\right)$ satisfying FDR criterion. In (a-h) and (l), an admissible super-cause node is $\boldsymbol{X}^{*}=\left\{ X\right\}$, an admissible super-mediator node is $\boldsymbol{M}^{*}=\left\{ M\right\}$ and an admissible super-effect node is $\boldsymbol{Y}^{*}=\left\{ Y\right\}$; in (i) and (j), an admissible super-cause node is $\boldsymbol{X}^{*}=\left\{ X\right\}$ and an admissible super-effect node is $\boldsymbol{Y}^{*}=\left\{ Y\right\}$, but the corresponding admissible super-mediator node $\boldsymbol{M}^{*}$ must be enlarged to $\{V, M \}$ and $\{U, M \}$, respectively; in (k), an admissible super-cause node needs enlargement to $\boldsymbol{X}^{*}=\left\{ X,U\right\}$, and the corresponding admissible super-mediator node is $\boldsymbol{M}^{*}=\left\{ M,V\right\}$ and an admissible super-effect node is $\boldsymbol{Y}^{*}=\left\{ Y\right\}$.
• Figure 3: Example ADMGs relative to the cause and effect of interest $X,Y$ not satisfying FDR criterion.
• Figure 4: Example ADMG that satisfies the FDR criterion, together with the modified graphs that appear in the do-calculus derivation of eq. \ref{['eq: interventional dist derivation Figure 2 (f)']}.

Theorems & Definitions (32)

• Definition 1: Front-door reducibility criterion, FDR criterion
• Theorem 1
• proof
• Theorem 2: Equivalence between FDR adjustment and FDR criterion satisfaction
• proof
• Definition 2: FDR triple
• Proposition 1: Effect minimality
• proof
• Theorem 3: Correctness of the FDR triple construction
• proof