# Nested Counterfactual Identification from Arbitrary Surrogate Experiments

arXiv: https://arxiv.org/abs/2107.03190 · HTML: https://mlsanity.com/paper/2107.03190/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 30 references that are in the map.
- First posted on arXiv 7 Jul 2021.

## Builds on (most cited first, 5 of 5)

- [Direct and Indirect Effects](https://mlsanity.com/paper/1301.2300/) (2013)
- [Probabilistic Evaluation of Sequential Plans from Causal Models with Hidden Variables](https://mlsanity.com/paper/1302.4977/) (2013)
- [A Probabilistic Calculus of Actions](https://mlsanity.com/paper/1302.6835/) (2013)
- [Testing Identifiability of Causal Effects](https://mlsanity.com/paper/1302.4948/) (2013)
- [Statistical Causality from a Decision-Theoretic Perspective](https://mlsanity.com/paper/1405.2292/) (2014)

## Built on by (most cited first, 30 of 38)

- [Partial Counterfactual Identification from Observational and Experimental Data](https://mlsanity.com/paper/2110.05690/) (2021)
- [Neural Causal Models for Counterfactual Identification and Estimation](https://mlsanity.com/paper/2210.00035/) (2022)
- [Counterfactual Identifiability of Bijective Causal Models](https://mlsanity.com/paper/2302.02228/) (2023)
- [Causal Effect Identification in Cluster DAGs](https://mlsanity.com/paper/2202.12263/) (2022)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Sharp Bounds for Generalized Causal Sensitivity Analysis](https://mlsanity.com/paper/2305.16988/) (2023)
- [Causal Inference Through the Structural Causal Marginal Problem](https://mlsanity.com/paper/2202.01300/) (2022)
- [Counterfactual Image Editing](https://mlsanity.com/paper/2403.09683/) (2024)
- [Counterfactual (Non-)identifiability of Learned Structural Causal Models](https://mlsanity.com/paper/2301.09031/) (2023)
- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [Partial Counterfactual Identification of Continuous Outcomes with a Curvature Sensitivity Model](https://mlsanity.com/paper/2306.01424/) (2023)
- [Finding Counterfactually Optimal Action Sequences in Continuous State Spaces](https://mlsanity.com/paper/2306.03929/) (2023)
- [Disentangling Causal Effects from Sets of Interventions in the Presence of Unobserved Confounders](https://mlsanity.com/paper/2210.05446/) (2022)
- [Causal Fairness under Unobserved Confounding: A Neural Sensitivity Framework](https://mlsanity.com/paper/2311.18460/) (2023)
- [Modular Learning of Deep Causal Generative Models for High-dimensional Causal Inference](https://mlsanity.com/paper/2401.01426/) (2024)
- [On the Complexity of Counterfactual Reasoning](https://mlsanity.com/paper/2211.13447/) (2022)
- [Counterfactual Identifiability via Dynamic Optimal Transport](https://mlsanity.com/paper/2510.08294/) (2025)
- [On Identifiability of Conditional Causal Effects](https://mlsanity.com/paper/2306.11755/) (2023)
- [Counterfactual Effect Decomposition in Multi-Agent Sequential Decision Making](https://mlsanity.com/paper/2410.12539/) (2024)
- [Partial counterfactual identification and uplift modeling: theoretical results and real-world assessment](https://mlsanity.com/paper/2211.07264/) (2022)
- [s-ID: Causal Effect Identification in a Sub-Population](https://mlsanity.com/paper/2309.02281/) (2023)
- [Path-specific effects for pulse-oximetry guided decisions in critical care](https://mlsanity.com/paper/2506.12371/) (2025)
- [Computational Identifiability](https://mlsanity.com/paper/2606.19361/) (2026)
- [Simulating counterfactuals](https://mlsanity.com/paper/2306.15328/) (2023)
- [Agent-Specific Effects: A Causal Effect Propagation Analysis in Multi-Agent MDPs](https://mlsanity.com/paper/2310.11334/) (2023)
- [Bayesian Networks for Causal Analysis in Socioecological Systems](https://mlsanity.com/paper/2401.10101/) (2024)
- [Causal Effect Identification in a Sub-Population with Latent Variables](https://mlsanity.com/paper/2405.14547/) (2024)
- [Teleporter Theory: A General and Simple Approach for Modeling Cross-World Counterfactual Causality](https://mlsanity.com/paper/2406.11501/) (2024)
- [Learning Counterfactual Outcomes Under Rank Preservation](https://mlsanity.com/paper/2502.06398/) (2025)
- [DISCO: Mitigating Bias in Deep Learning with Conditional Distance Correlation](https://mlsanity.com/paper/2506.11653/) (2025)

## Abstract (arXiv)

The Ladder of Causation describes three qualitatively different types of activities an agent may be interested in engaging in, namely, seeing (observational), doing (interventional), and imagining (counterfactual) (Pearl and Mackenzie, 2018). The inferential challenge imposed by the causal hierarchy is that data is collected by an agent observing or intervening in a system (layers 1 and 2), while its goal may be to understand what would have happened had it taken a different course of action, contrary to what factually ended up happening (layer 3). While there exists a solid understanding of the conditions under which cross-layer inferences are allowed from observations to interventions, the results are somewhat scarcer when targeting counterfactual quantities. In this paper, we study the identification of nested counterfactuals from an arbitrary combination of observations and experiments. Specifically, building on a more explicit definition of nested counterfactuals, we prove the counterfactual unnesting theorem (CUT), which allows one to map arbitrary nested counterfactuals to unnested ones. For instance, applications in mediation and fairness analysis usually evoke notions of direct, indirect, and spurious effects, which naturally require nesting. Second, we introduce a sufficient and necessary graphical condition for counterfactual identification from an arbitrary combination of observational and experimental distributions. Lastly, we develop an efficient and complete algorithm for identifying nested counterfactuals; failure of the algorithm returning an expression for a query implies it is not identifiable.