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Note on the identification of total effect in Cluster-DAGs with cycles

Clément Yvernes

TL;DR

This paper addresses the identifiability of total effects in cluster-DAGs when cycles may occur within clusters while the underlying micro-DAGs remain acyclic. It develops a complete graphical framework by introducing a minimal compatible graph and an unfolded representation, establishing a direct link between cluster-level d-separation and structure-of-interest connections across all compatible micrographs. A key result is that one only needs to consider clusters up to size $4$, with constructive procedures to move arrows and reduce complexity without affecting identifiability. The outcome is a sound and complete criterion for total-effect identifiability under do-calculus in the cluster-DAG setting, enabling practical causal inference with coarse-grained representations while handling cycles inside clusters.

Abstract

In this note, we discuss the identifiability of a total effect in cluster-DAGs, allowing for cycles within the cluster-DAG (while still assuming the associated underlying DAG to be acyclic). This is presented into two key results: first, restricting the cluster-DAG to clusters containing at most four nodes; second, adapting the notion of d-separation. We provide a graphical criterion to address the identifiability problem.

Note on the identification of total effect in Cluster-DAGs with cycles

TL;DR

This paper addresses the identifiability of total effects in cluster-DAGs when cycles may occur within clusters while the underlying micro-DAGs remain acyclic. It develops a complete graphical framework by introducing a minimal compatible graph and an unfolded representation, establishing a direct link between cluster-level d-separation and structure-of-interest connections across all compatible micrographs. A key result is that one only needs to consider clusters up to size , with constructive procedures to move arrows and reduce complexity without affecting identifiability. The outcome is a sound and complete criterion for total-effect identifiability under do-calculus in the cluster-DAG setting, enabling practical causal inference with coarse-grained representations while handling cycles inside clusters.

Abstract

In this note, we discuss the identifiability of a total effect in cluster-DAGs, allowing for cycles within the cluster-DAG (while still assuming the associated underlying DAG to be acyclic). This is presented into two key results: first, restricting the cluster-DAG to clusters containing at most four nodes; second, adapting the notion of d-separation. We provide a graphical criterion to address the identifiability problem.

Paper Structure

This paper contains 15 sections, 16 theorems, 3 equations, 4 figures.

Table of Contents

  1. Definitions and notations
  2. Notations.
  3. Active vertices and paths.
  4. Separated sets.
  5. Mutilated graph.
  6. Structural Causal Models and associated graphs.
  7. Interventions and do-operator.
  8. Main problem
  9. A useful characterisation of d-separation
  10. Basic Properties
  11. d-separation in C-DAGs
  12. Reducing the size of the cluster-DAG.
  13. A graphical criterion for d-separation in cluster-DAGs
  14. Solving Problem \ref{['pb:main_goal']}
  15. Conclusion

Key Result

Theorem 1

Let ${\mathcal{G}}$ be an ADMG. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be pairwise disjoint subsets of nodes of ${\mathcal{G}}$. The following properties are equivalent:

Figures (4)

  • Figure 1: Example of Cluster-DAG (left) and a micro-DAG compatible (right).
  • Figure 2: An ADMG in which there is a structure of interest in red that connects $X$ and $Y$ under $\{F, E\}$.
  • Figure 3: Example of Cluster-DAG (left) its minimal compatible graph (right).
  • Figure 4: 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 represents the corresponding unfolded graphs. The black arrows corresponds to ${{\mathcal{G}}^m_{\min}}$, whereas the red arrows represent the "to choose" 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, Lemma \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 $A_1 \in \text{Anc}(C_1, {{\mathcal{G}}^m})$.

Theorems & Definitions (41)

  • Definition 1: Cluster-DAG
  • Definition 2: Micro Graph Compatible
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1: d-connection with structures of interests
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 31 more