Relaxing partition admissibility in Cluster-DAGs: a causal calculus with arbitrary variable clustering

TL;DR

The paper extends Cluster-DAGs to allow cycles at the cluster level by relaxing partition admissibility, enabling causal reasoning across arbitrary variable clusterings. It introduces a structure-of-interest framework, along with canonical and unfolded graphs, to derive a sound and atomically complete calculus for cluster-level interventions, closely aligning with Pearl's do-calculus. A key practical insight is that any cluster can be reduced to size at most 3 without changing the calculus outcomes, making computation tractable even for large clusters. This work broadens the applicability of C-DAGs to intractable real-world settings while preserving a principled identification theory and offering strategies to manage unknown cluster sizes. Future directions include achieving global completeness and extending the approach to micro-level interventions when only the C-DAG is known.

Abstract

Cluster DAGs (C-DAGs) provide an abstraction of causal graphs in which nodes represent clusters of variables, and edges encode both cluster-level causal relationships and dependencies arisen from unobserved confounding. C-DAGs define an equivalence class of acyclic causal graphs that agree on cluster-level relationships, enabling causal reasoning at a higher level of abstraction. However, when the chosen clustering induces cycles in the resulting C-DAG, the partition is deemed inadmissible under conventional C-DAG semantics. In this work, we extend the C-DAG framework to support arbitrary variable clusterings by relaxing the partition admissibility constraint, thereby allowing cyclic C-DAG representations. We extend the notions of d-separation and causal calculus to this setting, significantly broadening the scope of causal reasoning across clusters and enabling the application of C-DAGs in previously intractable scenarios. Our calculus is both sound and atomically complete with respect to the do-calculus: all valid interventional queries at the cluster level can be derived using our rules, each corresponding to a primitive do-calculus step.

Figures (9)

• Figure 1: Left: a C-DAG ${{\mathcal{G}}^C} = (\mathcal{V}^C, \mathcal{E}^C)$, Right: a graph ${{\mathcal{G}}^m}=(\mathcal{V}^m, \mathcal{E}^m)$ that is compatible with ${{\mathcal{G}}^C}$. For example, ${}^2A \in \mathcal{V}^C$ corresponds to $\{A_1,A_2\} \subseteq \mathcal{V}^m$. In ${{\mathcal{G}}^m}$, a structure of interest (Definition \ref{['def:structure_of_interest']}) is highlighted in bold black, with all other nodes and edges shown in gray.
• Figure 2: On the first row (Figures \ref{['fig:example_gcu:1']}, \ref{['fig:example_gcu:2']} and \ref{['fig:example_gcu:3']}), three examples of C-DAG are given. On the second row (respectively, Figures \ref{['fig:example_gcu:4']}, \ref{['fig:example_gcu:5']} and \ref{['fig:example_gcu:6']}), we represent the corresponding unfolded and canonical compatible graphs. The plain and dashed arrows corresponds to ${{\mathcal{G}}^m_{\text{can}}}$, whereas the dotted arrows represent the "eligible" arrows. Lemma \ref{['lemma:gm_subgraph_gcu']} and Figure \ref{['fig:example_gcu:5']} show that there is no graph compatible with the C-DAG depicted in Figure \ref{['fig:example_gcu:2']} such that $A_1$ and $C_1$ are connected by a directed path. Similarly, Proposition \ref{['lemma:gm_cup_g3_compatible']} and Figure \ref{['fig:example_gcu:6']} show that there is no graph ${{\mathcal{G}}^m}$ compatible with the C-DAG depicted in Figure \ref{['fig:example_gcu:3']} such that $A_1 \in \text{Anc}(Z_1, {{\mathcal{G}}^m})$.
• Figure 3: Top: ${{\mathcal{G}}^C}$ (left) and ${\mathcal{G}}^C_{\leq 3}$ (right). Bottom: a graph compatible with ${{\mathcal{G}}^C}$ (left) and a graph compatible with ${\mathcal{G}}^C_{\leq 3}$ (right). In Figure \ref{['fig:example:1']}, the arrows in bold black represent a structure of interest that connects $X^m$ and $Y^m$ under $C^m \cup A^m$. In Figure \ref{['fig:example:2']}, the arrows in bold black represent the structure of interest that connects $X^m_{\leq 3}$ and $Y^m_{\leq 3}$ under $C^m_{\leq 3} \cup A^m_{\leq 3}$, which is obtained by applying the strategy used in the proof of Theorem \ref{['th:infinity_leq_three']}.
• Figure 4: Figure \ref{['fig:infty_geq_3:Gc']} depicts a cluster ${{\mathcal{G}}^C}$. Figure \ref{['fig:infty_geq_3:Gm']} illustrates a graph compatible with ${{\mathcal{G}}^C}$. Figure \ref{['fig:infty_geq_3:Gm_leq2']} illustrates the unique graph compatible with ${\mathcal{G}}^C_{\leq2}$.
• Figure 5: We represent the three forms that arrows can take around a vertex with multiple arrow in a structure of interest. Some vertices have incoming arrows (left), some have incoming arrows and a single outgoing arrow (middle), and some have exactly two outgoing arrows and no incoming arrows (right).

Theorems & Definitions (56)

• Definition 1: Cluster-DAG
• Definition 2: Class of Compatible Graphs
• Definition 3
• Definition 4: Connecting structure of interest
• Example 1
• Theorem 1: D-connection with structures of interests
• Definition 5: Canonical Compatible Graph
• Proposition 1
• Definition 6: Unfolded graph
• Proposition 2