# Interventional Causal Discovery in a Mixture of DAGs

arXiv: https://arxiv.org/abs/2406.08666 · HTML: https://mlsanity.com/paper/2406.08666/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 53 references that are in the map.
- First posted on arXiv 12 Jun 2024.

## Builds on (most cited first, 22 of 22)

- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [Causal Discovery from Heterogeneous/Nonstationary Data with Independent Changes](https://mlsanity.com/paper/1903.01672/) (2019)
- [Characterizing and Learning Equivalence Classes of Causal DAGs under Interventions](https://mlsanity.com/paper/1802.06310/) (2018)
- [Permutation-Based Causal Structure Learning with Unknown Intervention Targets](https://mlsanity.com/paper/1910.09007/) (2019)
- [Learning Causal Graphs with Small Interventions](https://mlsanity.com/paper/1511.00041/) (2015)
- [Cost-Optimal Learning of Causal Graphs](https://mlsanity.com/paper/1703.02645/) (2017)
- [Active Structure Learning of Causal DAGs via Directed Clique Tree](https://mlsanity.com/paper/2011.00641/) (2020)
- [Constraint-Based Causal Discovery using Partial Ancestral Graphs in the presence of Cycles](https://mlsanity.com/paper/2005.00610/) (2020)
- [Directed Cyclic Graphical Representations of Feedback Models](https://mlsanity.com/paper/1302.4982/) (2013)
- [Experimental Design for Cost-Aware Learning of Causal Graphs](https://mlsanity.com/paper/1810.11867/) (2018)
- [Causal Structure Discovery from Distributions Arising from Mixtures of DAGs](https://mlsanity.com/paper/2001.11940/) (2020)
- [Efficient Intervention Design for Causal Discovery with Latents](https://mlsanity.com/paper/2005.11736/) (2020)
- [Verification and search algorithms for causal DAGs](https://mlsanity.com/paper/2206.15374/) (2022)
- [An Algorithm for Deciding if a Set of Observed Independencies Has a Causal Explanation](https://mlsanity.com/paper/1303.5435/) (2013)
- [Learning Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2201.11211/) (2022)
- [Learning Mixtures of DAG Models](https://mlsanity.com/paper/1301.7415/) (2013)
- [A Distance Covariance-based Kernel for Nonlinear Causal Clustering in Heterogeneous Populations](https://mlsanity.com/paper/2106.03480/) (2021)
- [Discovering Mixtures of Structural Causal Models from Time Series Data](https://mlsanity.com/paper/2310.06312/) (2023)
- [Intervention Efficient Algorithms for Approximate Learning of Causal Graphs](https://mlsanity.com/paper/2012.13976/) (2020)
- [A Unified Experiment Design Approach for Cyclic and Acyclic Causal Models](https://mlsanity.com/paper/2205.10083/) (2022)
- [Disentangling Mixtures of Unknown Causal Interventions](https://mlsanity.com/paper/2210.03242/) (2022)
- [CCSL: A Causal Structure Learning Method from Multiple Unknown Environments](https://mlsanity.com/paper/2111.09666/) (2021)

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

- [Covariate Dependent Mixture of Bayesian Networks](https://mlsanity.com/paper/2501.05745/) (2025)
- [Interpretable Clustering with Adaptive Heterogeneous Causal Structure Learning in Mixed Observational Data](https://mlsanity.com/paper/2509.04415/) (2025)
- [Causal Discovery Should Embrace the Wisdom of the Crowd](https://mlsanity.com/paper/2603.02678/) (2026)

## Abstract (arXiv)

Causal interactions among a group of variables are often modeled by a single causal graph. In some domains, however, these interactions are best described by multiple co-existing causal graphs, e.g., in dynamical systems or genomics. This paper addresses the hitherto unknown role of interventions in learning causal interactions among variables governed by a mixture of causal systems, each modeled by one directed acyclic graph (DAG). Causal discovery from mixtures is fundamentally more challenging than single-DAG causal discovery. Two major difficulties stem from (i)~an inherent uncertainty about the skeletons of the component DAGs that constitute the mixture and (ii)~possibly cyclic relationships across these component DAGs. This paper addresses these challenges and aims to identify edges that exist in at least one component DAG of the mixture, referred to as the true edges. First, it establishes matching necessary and sufficient conditions on the size of interventions required to identify the true edges. Next, guided by the necessity results, an adaptive algorithm is designed that learns all true edges using O(n2) interventions, where n is the number of nodes. Remarkably, the size of the interventions is optimal if the underlying mixture model does not contain cycles across its components. More generally, the gap between the intervention size used by the algorithm and the optimal size is quantified. It is shown to be bounded by the cyclic complexity number of the mixture model, defined as the size of the minimal intervention that can break the cycles in the mixture, which is upper bounded by the number of cycles among the ancestors of a node.