s-Separation in Summary DAGs: Sound and Complete CI Identification

Summary

s-Separation extends d-separation to summary causal DAGs. A CI statement $X\perp \perp Y\mid Z$ holds in a summary DAG $(\mathcal{H},f)$ if $X$ and $Y$ are d-separated in every causal DAG compatible with $\mathcal{H}$. This is equivalently characterized by d-separation in the canonical causal DAG ${\mathcal{G}}_{\mathcal{H}}$. Theorem 4.2 establishes that s-separation is sound and complete: the s-separation algorithm correctly identifies exactly those CIs that are valid across all compatible DAGs.

Overview

In a standard causal DAG, d-separation identifies all conditional independence relationships encoded in the graph. For a summary DAG — which represents a set of compatible causal DAGs — we need a stricter notion: a CI should only be claimed if it holds in all compatible DAGs (conservative inference). s-Separation provides this.

Main Content

CI Validity in Summary DAGs

Valid CI in a Summary Causal DAG (Definition 6)

A CI statement $(X\perp \perp Y\mid Z)$ is valid in a summary causal DAG $(\mathcal{H},f)$ if and only if it holds in every causal DAG $\mathcal{G}\in \{{\mathcal{G}}_i{\}}_{\mathcal{H}}$ (every DAG compatible with $\mathcal{H}$).

Equivalently: $X$ and $Y$ are d-separated given $Z$ in every compatible $\mathcal{G}$.

This is stricter than d-separation in $\mathcal{H}$ alone, because $\mathcal{H}$ might encode a CI that not all compatible DAGs share.

s-Separation

s-Separation (Definition 7)

Given a summary causal DAG $(\mathcal{H},f)$ and disjoint sets $X,Y,Z\subseteq \mathcal{V}(\mathcal{H})$, nodes $X$ and $Y$ are s-separated given $Z$ in $(\mathcal{H},f)$, denoted:

$$X\perp {\perp }_{s}Y\mid Z_{(\mathcal{H},f)}$$

if and only if $f^{-1}(X)$ and $f^{-1}(Y)$ are d-separated given $f^{-1}(Z)$ in the canonical causal DAG ${\mathcal{G}}_{\mathcal{H}}$.

That is: expand the summary DAG nodes back to their original variable sets, then apply standard d-separation in the canonical DAG.

Key property: Because ${\mathcal{G}}_{\mathcal{H}}$ is a supergraph of any compatible $\mathcal{G}$ (it has the most edges), d-separation in ${\mathcal{G}}_{\mathcal{H}}$ is the strictest criterion — it identifies only CIs present in all compatible DAGs, not just some.

Algorithm for s-Separation

s-Separation Algorithm

Given summary DAG $(\mathcal{H},f)$:

1. Construct the canonical causal DAG ${\mathcal{G}}_{\mathcal{H}}$ (using Definition 5 in Canonical Causal DAGs).
2. For query $(X\perp \perp Y\mid Z)$ in the summary: expand to $(f^{-1}(X)\perp \perp f^{-1}(Y)\mid f^{-1}(Z))$ in ${\mathcal{G}}_{\mathcal{H}}$.
3. Apply any standard d-separation algorithm on ${\mathcal{G}}_{\mathcal{H}}$.

Result: the CI holds in the summary DAG iff it holds in this expanded query.

Why naive d-separation on $\mathcal{H}$ fails: Consider a 5-node summary. Two nodes $A$ and $C$ may appear d-separated in $\mathcal{H}$, but in the canonical DAG the within-cluster edges create a path between $f^{-1}(A)$ and $f^{-1}(C)$ — so the CI does not actually hold in all compatible DAGs.

s-Separation Example (from §4.2.1)

Referring to Fig. 3, consider summary ${\mathcal{H}}_1$ (Fig. 4a, contracting $B$ and $C$ into cluster $BC$).

Query: $(B\perp {\perp }_{d}E\mid D)$ in ${\mathcal{H}}_1$ — does this hold?

In the canonical DAG ${\mathcal{G}}_{{\mathcal{H}}_1}$: the cluster $BC$ is expanded; check d-separation of $f^{-1}(B)=\{B,C\}$ and $f^{-1}(E)=\{E\}$ given $f^{-1}(D)=\{D\}$ in ${\mathcal{G}}_{{\mathcal{H}}_1}$.

Result: $(B\perp \perp E\mid D)$ and $(C\perp \perp E\mid D)$ hold in ${\mathcal{H}}_1$ (established in the paper), but $(BC\perp \perp E\mid D)$ with the canonical expansion shows this holds only if the within-cluster $B\to C$ edge doesn’t create a path — which must be checked explicitly.

Soundness and Completeness

Theorem 4.2 — Soundness and Completeness of s-Separation

Let $(\mathcal{H},f)$ be a summary causal DAG for $\mathcal{G}$, and let $X,Y,Z\subseteq \mathcal{V}(\mathcal{H})$ be disjoint sets.

$X$ and $Y$ are s-separated given $Z$ in $(\mathcal{H},f)$ if and only if $X$ and $Y$ are d-separated given $Z$ in every causal DAG $\mathcal{G}\in \{{\mathcal{G}}_i{\}}_{\mathcal{H}}$ compatible with $\mathcal{H}$.

Formally:

$$(X\perp {\perp }_{s}Y\mid Z{)}_{(\mathcal{H},f)}⟺\forall \mathcal{G}\in \{{\mathcal{G}}_i{\}}_{\mathcal{H}}:(f^{-1}(X)\perp {\perp }_d{f}^{-1}(Y)\mid f^{-1}(Z){)}_{\mathcal{G}}$$

Soundness: If s-separation says $X\perp \perp Y\mid Z$, then this CI holds in all compatible DAGs (no false claims of independence).

Completeness: If $X\perp \perp Y\mid Z$ holds in all compatible DAGs, s-separation will identify it (no missed valid CIs).

Proof: Uses the equivalence of RBs (Theorem 4.1): the canonical DAG ${\mathcal{G}}_{\mathcal{H}}$ is compatible with $\mathcal{H}$ and is a supergraph of any other compatible DAG. Therefore d-separation in ${\mathcal{G}}_{\mathcal{H}}$ is equivalent to d-separation in all compatible DAGs.

Practical Implication

s-Separation provides a conservative but correct inference tool:

• It may identify fewer CIs than the true original DAG (because it uses the strictest compatible DAG).
• But every CI it identifies is guaranteed to hold — no spurious independence assumptions.
• This is the correct tradeoff for summarization: we lose some precision but never introduce incorrect assumptions into a causal analysis.

Connections

• Extends Directed Acyclic Graphs’s d-separation to the summary context.
• The soundness guarantee is critical for Do-Calculus in Summary Causal DAGs — do-calculus requires valid CI statements as its core primitive.
• The conservative nature parallels the approach in Frequentist Causal Estimation — using larger adjustment sets (more confounders) is conservative but unbiased.

See Also

• Canonical Causal DAGs — provides the mathematical basis for s-separation
• Summary Causal DAGs — the object s-separation operates on
• Do-Calculus in Summary Causal DAGs — uses s-separation for causal effect identification
• Directed Acyclic Graphs — foundational d-separation from which s-separation is built