# Counterfactual Identifiability of Bijective Causal Models

arXiv: https://arxiv.org/abs/2302.02228 · HTML: https://mlsanity.com/paper/2302.02228/

## Key facts

- Cited by 43 papers in the map (data through 14 Oct 2026).
- Builds on 67 of its 127 references that are in the map.
- First posted on arXiv 4 Feb 2023.

## Builds on (most cited first, 30 of 67)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [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)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Underspecification Presents Challenges for Credibility in Modern Machine Learning](https://mlsanity.com/paper/2011.03395/) (2020)
- [Causal Effect Inference with Deep Latent-Variable Models](https://mlsanity.com/paper/1705.08821/) (2017)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Learning Representations for Counterfactual Inference](https://mlsanity.com/paper/1605.03661/) (2016)
- [Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning](https://mlsanity.com/paper/1805.08651/) (2018)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [AttGAN: Facial Attribute Editing by Only Changing What You Want](https://mlsanity.com/paper/1711.10678/) (2017)
- [Deep Structural Causal Models for Tractable Counterfactual Inference](https://mlsanity.com/paper/2006.06485/) (2020)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [CausalGAN: Learning Causal Implicit Generative Models with Adversarial Training](https://mlsanity.com/paper/1709.02023/) (2017)
- [Identifying Causal Effects With Proxy Variables of an Unmeasured Confounder](https://mlsanity.com/paper/1609.08816/) (2016)
- [From Variational to Deterministic Autoencoders](https://mlsanity.com/paper/1903.12436/) (2019)
- [Learning in Implicit Generative Models](https://mlsanity.com/paper/1610.03483/) (2016)
- [Fader Networks: Manipulating Images by Sliding Attributes](https://mlsanity.com/paper/1706.00409/) (2017)
- [Counterfactual Off-Policy Evaluation with Gumbel-Max Structural Causal Models](https://mlsanity.com/paper/1905.05824/) (2019)
- [The Blessings of Multiple Causes](https://mlsanity.com/paper/1805.06826/) (2018)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)

## Built on by (most cited first, 30 of 43)

- [Causal normalizing flows: from theory to practice](https://mlsanity.com/paper/2306.05415/) (2023)
- [From Identifiable Causal Representations to Controllable Counterfactual Generation: A Survey on Causal Generative Modeling](https://mlsanity.com/paper/2310.11011/) (2023)
- [Counterfactual Image Editing](https://mlsanity.com/paper/2403.09683/) (2024)
- [Learning Causally Disentangled Representations via the Principle of Independent Causal Mechanisms](https://mlsanity.com/paper/2306.01213/) (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)
- [Deep Backtracking Counterfactuals for Causally Compliant Explanations](https://mlsanity.com/paper/2310.07665/) (2023)
- [Learning Structural Causal Models through Deep Generative Models: Methods, Guarantees, and Challenges](https://mlsanity.com/paper/2405.05025/) (2024)
- [Counterfactual Fairness by Combining Factual and Counterfactual Predictions](https://mlsanity.com/paper/2409.01977/) (2024)
- [DeCaFlow: A deconfounding causal generative model](https://mlsanity.com/paper/2503.15114/) (2025)
- [Deep Learning With DAGs](https://mlsanity.com/paper/2401.06864/) (2024)
- [Distinguishing Cause from Effect with Causal Velocity Models](https://mlsanity.com/paper/2502.05122/) (2025)
- [Counterfactual Identifiability via Dynamic Optimal Transport](https://mlsanity.com/paper/2510.08294/) (2025)
- [Towards Characterizing Domain Counterfactuals For Invertible Latent Causal Models](https://mlsanity.com/paper/2306.11281/) (2023)
- [Consistent End-to-End Estimation for Counterfactual Fairness](https://mlsanity.com/paper/2310.17687/) (2023)
- [Interpretable Neural Causal Models with TRAM-DAGs](https://mlsanity.com/paper/2503.16206/) (2025)
- [Multi-Domain Causal Discovery in Bijective Causal Models](https://mlsanity.com/paper/2504.21261/) (2025)
- [Counterfactual reasoning: an analysis of in-context emergence](https://mlsanity.com/paper/2506.05188/) (2025)
- [Causal-Adapter: Taming Text-to-Image Diffusion for Faithful Counterfactual Generation](https://mlsanity.com/paper/2509.24798/) (2025)
- [DoFlow: Flow-based Generative Models for Interventional and Counterfactual Forecasting on Time Series](https://mlsanity.com/paper/2511.02137/) (2025)
- [Causal Bayesian Optimization via Exogenous Distribution Learning](https://mlsanity.com/paper/2402.02277/) (2024)
- [Flow-based Generative Modeling of Potential Outcomes and Counterfactuals](https://mlsanity.com/paper/2505.16051/) (2025)
- [FoundCause: Causal Discovery with Latent Confounders from Observational Data](https://mlsanity.com/paper/2606.17516/) (2026)
- [Lookahead Counterfactual Fairness](https://mlsanity.com/paper/2412.01065/) (2024)
- [Counterfactually Fair Conformal Prediction](https://mlsanity.com/paper/2510.08724/) (2025)
- [Advancing Counterfactual Inference through Nonlinear Quantile Regression](https://mlsanity.com/paper/2306.05751/) (2023)
- [Agent-Specific Effects: A Causal Effect Propagation Analysis in Multi-Agent MDPs](https://mlsanity.com/paper/2310.11334/) (2023)
- [A New Approach to Backtracking Counterfactual Explanations: A Unified Causal Framework for Efficient Model Interpretability](https://mlsanity.com/paper/2505.02435/) (2025)
- [Flow IV: Counterfactual Inference In Nonseparable Outcome Models Using Instrumental Variables](https://mlsanity.com/paper/2508.01321/) (2025)
- [CEPAE: Conditional Entropy-Penalized Autoencoders for Time Series Counterfactuals](https://mlsanity.com/paper/2602.15546/) (2026)

## Abstract (arXiv)

We study counterfactual identifiability in causal models with bijective generation mechanisms (BGM), a class that generalizes several widely-used causal models in the literature. We establish their counterfactual identifiability for three common causal structures with unobserved confounding, and propose a practical learning method that casts learning a BGM as structured generative modeling. Learned BGMs enable efficient counterfactual estimation and can be obtained using a variety of deep conditional generative models. We evaluate our techniques in a visual task and demonstrate its application in a real-world video streaming simulation task.