# Counterfactual (Non-)identifiability of Learned Structural Causal Models

arXiv: https://arxiv.org/abs/2301.09031 · HTML: https://mlsanity.com/paper/2301.09031/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 67 references that are in the map.
- First posted on arXiv 22 Jan 2023.

## Builds on (most cited first, 30 of 39)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Counterfactual Fairness](https://mlsanity.com/paper/1703.06856/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [Adversarially Learned Inference](https://mlsanity.com/paper/1606.00704/) (2016)
- [Deep Structural Causal Models for Tractable Counterfactual Inference](https://mlsanity.com/paper/2006.06485/) (2020)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Learning in Implicit Generative Models](https://mlsanity.com/paper/1610.03483/) (2016)
- [Counterfactual Off-Policy Evaluation with Gumbel-Max Structural Causal Models](https://mlsanity.com/paper/1905.05824/) (2019)
- [The Causal-Neural Connection: Expressiveness, Learnability, and Inference](https://mlsanity.com/paper/2107.00793/) (2021)
- [Deep End-to-end Causal Inference](https://mlsanity.com/paper/2202.02195/) (2022)
- [Diffusion Causal Models for Counterfactual Estimation](https://mlsanity.com/paper/2202.10166/) (2022)
- [Causal structure based root cause analysis of outliers](https://mlsanity.com/paper/1912.02724/) (2019)
- [Counterfactual Credit Assignment in Model-Free Reinforcement Learning](https://mlsanity.com/paper/2011.09464/) (2020)
- [Partial Counterfactual Identification from Observational and Experimental Data](https://mlsanity.com/paper/2110.05690/) (2021)
- [Sample-Efficient Reinforcement Learning via Counterfactual-Based Data Augmentation](https://mlsanity.com/paper/2012.09092/) (2020)
- [Evaluating and Mitigating Bias in Image Classifiers: A Causal Perspective Using Counterfactuals](https://mlsanity.com/paper/2009.08270/) (2020)
- [Explaining Black-Box Algorithms Using Probabilistic Contrastive Counterfactuals](https://mlsanity.com/paper/2103.11972/) (2021)

## Built on by (most cited first, 17 of 17)

- [Counterfactual Identifiability of Bijective Causal Models](https://mlsanity.com/paper/2302.02228/) (2023)
- [Modeling Causal Mechanisms with Diffusion Models for Interventional and Counterfactual Queries](https://mlsanity.com/paper/2302.00860/) (2023)
- [Counterfactual Image Editing](https://mlsanity.com/paper/2403.09683/) (2024)
- [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)
- [Emerging Synergies in Causality and Deep Generative Models: A Survey](https://mlsanity.com/paper/2301.12351/) (2023)
- [Position: Causal Machine Learning Requires Rigorous Synthetic Experiments for Broader Adoption](https://mlsanity.com/paper/2508.08883/) (2025)
- [Counterfactual Identifiability via Dynamic Optimal Transport](https://mlsanity.com/paper/2510.08294/) (2025)
- [A Definition and Roadmap for World Models](https://mlsanity.com/paper/2607.06401/) (2026)
- [Causal Bayesian Optimization via Exogenous Distribution Learning](https://mlsanity.com/paper/2402.02277/) (2024)
- [Robust Counterfactual Inference in Markov Decision Processes](https://mlsanity.com/paper/2502.13731/) (2025)
- [Causality-aligned Prompt Learning via Diffusion-based Counterfactual Generation](https://mlsanity.com/paper/2507.19882/) (2025)
- [Advancing Counterfactual Inference through Nonlinear Quantile Regression](https://mlsanity.com/paper/2306.05751/) (2023)
- [CEPAE: Conditional Entropy-Penalized Autoencoders for Time Series Counterfactuals](https://mlsanity.com/paper/2602.15546/) (2026)
- [Unsupervised Structural-Counterfactual Generation under Domain Shift](https://mlsanity.com/paper/2502.12013/) (2025)
- [Partially Functional Dynamic Backdoor Diffusion-based Causal Model](https://mlsanity.com/paper/2509.00472/) (2025)
- [CAFE: Counterfactual Prediction via Fast Posterior Estimation](https://mlsanity.com/paper/2609.32167/) (2026)

## Abstract (arXiv)

Recent advances in probabilistic generative modeling have motivated learning Structural Causal Models (SCM) from observational datasets using deep conditional generative models, also known as Deep Structural Causal Models (DSCM). If successful, DSCMs can be utilized for causal estimation tasks, e.g., for answering counterfactual queries. In this work, we warn practitioners about non-identifiability of counterfactual inference from observational data, even in the absence of unobserved confounding and assuming known causal structure. We prove counterfactual identifiability of monotonic generation mechanisms with single dimensional exogenous variables. For general generation mechanisms with multi-dimensional exogenous variables, we provide an impossibility result for counterfactual identifiability, motivating the need for parametric assumptions. As a practical approach, we propose a method for estimating worst-case errors of learned DSCMs' counterfactual predictions. The size of this error can be an essential metric for deciding whether or not DSCMs are a viable approach for counterfactual inference in a specific problem setting. In evaluation, our method confirms negligible counterfactual errors for an identifiable SCM from prior work, and also provides informative error bounds on counterfactual errors for a non-identifiable synthetic SCM.