Complete Causal Identification from Ancestral Graphs under Selection Bias

Abstract

Many causal discovery algorithms, including the celebrated FCI algorithm, output a Partial Ancestral Graph (PAG). PAGs serve as an abstract graphical representation of the underlying causal structure, modeled by directed acyclic graphs with latent and selection variables. This paper develops a characterization of the set of extended-type conditional independence relations that are invariant across all causal models represented by a PAG. This theory allows us to formulate a general measure-theoretic version of Pearl's causal calculus and a sound and complete identification algorithm for PAGs under selection bias. Our results also apply when PAGs are learned by certain algorithms that integrate observational data with experimental data and incorporate background knowledge.

Figures (14)

• Figure 1: An ADMG $\mathfrak{A}$ and the derived graphs $\mathfrak{A}_{\mathsf{do}(I_a)}$ and $\mathfrak{A}_{\mathsf{do}(I_a,b)}$ in \ref{['ex:admg']}.
• Figure 2: Two isADMGs $\mathfrak{A}^1$ and $\mathfrak{A}^2$ in \ref{['ex:mag_trans']} are the identical up to node type, whereas their MAG representations are different.
• Figure 3: Example about zhang2008causal in \ref{['rem:zhang']}: $\mathfrak{A}^1$ and $\mathfrak{A}^2$ are represented by $\mathfrak{M}$.
• Figure 4: \ref{['ex:countex_ci']} showing the failure of $\mathsf{IM}(\mathfrak{M}_{\mathsf{do}(I_a)})=\mathsf{IM}(\mathfrak{A}_{\mathsf{do}(I_a)}\mid \mathcal{S}_\mathfrak{A})$.
• Figure 5: MAG $\mathfrak{M}$ in \ref{['ex:dvsd*']} such that $d\underset{\mathfrak{M}_{\mathsf{do}(I_a)}}{\stackrel{\mathsf{d}}{\not\perp}} c\mid a$, but for $\mathfrak{A}^1,\mathfrak{A}^2\in [\mathfrak{M}]_\mathsf{G}$, we have $d\underset{\mathfrak{A}^1_{\mathsf{do}(I_a)}}{\stackrel{\mathsf{d}}{\perp}} c\mid \{a\}\cup \{s\}$ and $d\underset{\mathfrak{A}^2_{\mathsf{do}(I_a)}}{\stackrel{\mathsf{d}}{\perp}} c\mid \{a\}\cup \{s\}$. We have $d\underset{\mathfrak{A}^3_{\mathsf{do}(I_a)}}{\stackrel{\mathsf{d}}{\not\perp}} c\mid \{a\}\cup \{s\}$, but $\mathfrak{A}^3\notin [\mathfrak{M}]_\mathsf{G}$.

Theorems & Definitions (185)

• Definition 1.2: Causal relation (graphical version)
• Definition 1.3: Causal relation (conditional independence version)
• Example 1.4
• Definition 2.1: Causal ilsADMG
• Remark 2.2: Causal interpretation of isADMG
• Definition 2.3: Inducing path/walk
• Definition 2.4: Partial ancestral graphs with inputs (iPAGs)
• Remark 2.5
• Definition 2.6: Potentially directed/anterior paths and possible graphical relations
• Definition 2.7: Maximal ancestral graphs with inputs (iMAGs)