# Identifying Macro Causal Effects in C-DMGs over DMGs

arXiv: https://arxiv.org/abs/2506.19650 · HTML: https://mlsanity.com/paper/2506.19650/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 30 references that are in the map.
- First posted on arXiv 24 Jun 2025.

## Builds on (most cited first, 14 of 14)

- [Causal Effect Identification in Cluster DAGs](https://mlsanity.com/paper/2202.12263/) (2022)
- [Constraint-based Causal Discovery for Non-Linear Structural Causal Models with Cycles and Latent Confounders](https://mlsanity.com/paper/1807.03024/) (2018)
- [Markov Properties for Graphical Models with Cycles and Latent Variables](https://mlsanity.com/paper/1710.08775/) (2017)
- [Root Cause Identification for Collective Anomalies in Time Series given an Acyclic Summary Causal Graph with Loops](https://mlsanity.com/paper/2303.04038/) (2023)
- [Foundations of Causal Discovery on Groups of Variables](https://mlsanity.com/paper/2306.07047/) (2023)
- [Causal Calculus in the Presence of Cycles, Latent Confounders and Selection Bias](https://mlsanity.com/paper/1901.00433/) (2019)
- [Identifiability of total effects from abstractions of time series causal graphs](https://mlsanity.com/paper/2310.14691/) (2023)
- [Clustering and Structural Robustness in Causal Diagrams](https://mlsanity.com/paper/2111.04513/) (2021)
- [Identifiability of Direct Effects from Summary Causal Graphs](https://mlsanity.com/paper/2306.16958/) (2023)
- [A generalized back-door criterion](https://mlsanity.com/paper/1307.5636/) (2013)
- [Identifying Macro Conditional Independencies and Macro Total Effects in Summary Causal Graphs with Latent Confounding](https://mlsanity.com/paper/2407.07934/) (2024)
- [Towards identifiability of micro total effects in summary causal graphs with latent confounding: extension of the front-door criterion](https://mlsanity.com/paper/2406.05805/) (2024)
- [Identifying Macro Causal Effects in a C-DMG over ADMGs](https://mlsanity.com/paper/2504.01551/) (2025)
- [Note on the identification of total effect in Cluster-DAGs with cycles](https://mlsanity.com/paper/2504.07921/) (2025)

## Built on by (most cited first, 3 of 3)

- [Improving the Generation and Evaluation of Synthetic Data for Downstream Medical Causal Inference](https://mlsanity.com/paper/2510.18768/) (2025)
- [From Local to Cluster: A Unified Framework for Causal Discovery with Latent Variables](https://mlsanity.com/paper/2604.22416/) (2026)
- [Invariance of Clustering Operations in Causal Effect Identification](https://mlsanity.com/paper/2610.03101/) (2026)

## Abstract (arXiv)

The do-calculus is a sound and complete tool for identifying causal effects in acyclic directed mixed graphs (ADMGs) induced by structural causal models (SCMs). However, in many real-world applications, especially in high-dimensional setting, constructing a fully specified ADMG is often infeasible. This limitation has led to growing interest in partially specified causal representations, particularly through cluster-directed mixed graphs (C-DMGs), which group variables into clusters and offer a more abstract yet practical view of causal dependencies. While these representations can include cycles, recent work has shown that the do-calculus remains sound and complete for identifying macro-level causal effects in C-DMGs over ADMGs under the assumption that all clusters size are greater than 1. Nevertheless, real-world systems often exhibit cyclic causal dynamics at the structural level. To account for this, input-output structural causal models (ioSCMs) have been introduced as a generalization of SCMs that allow for cycles. ioSCMs induce another type of graph structure known as a directed mixed graph (DMG). Analogous to the ADMG setting, one can define C-DMGs over DMGs as high-level representations of causal relations among clusters of variables. In this paper, we prove that, unlike in the ADMG setting, the do-calculus is unconditionally sound and complete for identifying macro causal effects in C-DMGs over DMGs. Furthermore, we show that the graphical criteria for non-identifiability of macro causal effects previously established C-DMGs over ADMGs naturally extends to a subset of C-DMGs over DMGs.