# Transport-based Counterfactual Models

arXiv: https://arxiv.org/abs/2108.13025 · HTML: https://mlsanity.com/paper/2108.13025/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 78 references that are in the map.
- First posted on arXiv 30 Aug 2021.

## Builds on (most cited first, 29 of 29)

- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [DeepFool: a simple and accurate method to fool deep neural networks](https://mlsanity.com/paper/1511.04599/) (2015)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Counterfactual Fairness](https://mlsanity.com/paper/1703.06856/) (2017)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Algorithmic Recourse: from Counterfactual Explanations to Interventions](https://mlsanity.com/paper/2002.06278/) (2020)
- [FACE: Feasible and Actionable Counterfactual Explanations](https://mlsanity.com/paper/1909.09369/) (2019)
- [Path-Specific Counterfactual Fairness](https://mlsanity.com/paper/1802.08139/) (2018)
- [Fair Inference On Outcomes](https://mlsanity.com/paper/1705.10378/) (2017)
- [Preserving Causal Constraints in Counterfactual Explanations for Machine Learning Classifiers](https://mlsanity.com/paper/1912.03277/) (2019)
- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Towards Realistic Individual Recourse and Actionable Explanations in Black-Box Decision Making Systems](https://mlsanity.com/paper/1907.09615/) (2019)
- [Anchor regression: heterogeneous data meets causality](https://mlsanity.com/paper/1801.06229/) (2018)
- [Counterfactual Explanations Can Be Manipulated](https://mlsanity.com/paper/2106.02666/) (2021)
- [Fair Regression with Wasserstein Barycenters](https://mlsanity.com/paper/2006.07286/) (2020)
- [Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation](https://mlsanity.com/paper/2010.05862/) (2020)
- [Robust Counterfactual Explanations on Graph Neural Networks](https://mlsanity.com/paper/2107.04086/) (2021)
- [On the Fairness of Causal Algorithmic Recourse](https://mlsanity.com/paper/2010.06529/) (2020)
- [Optimal Transport for Multi-source Domain Adaptation under Target Shift](https://mlsanity.com/paper/1803.04899/) (2018)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [FlipTest: Fairness Testing via Optimal Transport](https://mlsanity.com/paper/1906.09218/) (2019)
- [The Sensitivity of Counterfactual Fairness to Unmeasured Confounding](https://mlsanity.com/paper/1907.01040/) (2019)
- [A survey of bias in Machine Learning through the prism of Statistical Parity for the Adult Data Set](https://mlsanity.com/paper/2003.14263/) (2020)
- [Fair Data Adaptation with Quantile Preservation](https://mlsanity.com/paper/1911.06685/) (2019)
- [Tackling Algorithmic Bias in Neural-Network Classifiers using Wasserstein-2 Regularization](https://mlsanity.com/paper/1908.05783/) (2019)
- [CARE: Coherent Actionable Recourse based on Sound Counterfactual Explanations](https://mlsanity.com/paper/2108.08197/) (2021)
- [An Optimal Transport Approach to Estimating Causal Effects via Nonlinear Difference-in-Differences](https://mlsanity.com/paper/2108.05858/) (2021)
- [Center-outward quantiles and the measurement of multivariate risk](https://mlsanity.com/paper/1912.04924/) (2019)

## Built on by (most cited first, 27 of 27)

- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [Partial Counterfactual Identification of Continuous Outcomes with a Curvature Sensitivity Model](https://mlsanity.com/paper/2306.01424/) (2023)
- [On the explainable properties of 1-Lipschitz Neural Networks: An Optimal Transport Perspective](https://mlsanity.com/paper/2206.06854/) (2022)
- [Towards the Causal Complete Cause of Multi-Modal Representation Learning](https://mlsanity.com/paper/2407.14058/) (2024)
- [GAN Estimation of Lipschitz Optimal Transport Maps](https://mlsanity.com/paper/2202.07965/) (2022)
- [Quantile-constrained Wasserstein projections for robust interpretability of numerical and machine learning models](https://mlsanity.com/paper/2209.11539/) (2022)
- [Sequential Conditional Transport on Probabilistic Graphs for Interpretable Counterfactual Fairness](https://mlsanity.com/paper/2408.03425/) (2024)
- [A survey of Identification and mitigation of Machine Learning algorithmic biases in Image Analysis](https://mlsanity.com/paper/2210.04491/) (2022)
- [Lookahead Counterfactual Fairness](https://mlsanity.com/paper/2412.01065/) (2024)
- [Learning with Differentially Private (Sliced) Wasserstein Gradients](https://mlsanity.com/paper/2502.01701/) (2025)
- [A primer on optimal transport for causal inference with observational data](https://mlsanity.com/paper/2503.07811/) (2025)
- [Iterative Partial Fulfillment of Counterfactual Explanations: Benefits and Risks](https://mlsanity.com/paper/2303.11111/) (2023)
- [Advancing Counterfactual Inference through Nonlinear Quantile Regression](https://mlsanity.com/paper/2306.05751/) (2023)
- [Measuring and Mitigating Biases in Motor Insurance Pricing](https://mlsanity.com/paper/2311.11900/) (2023)
- [Fair Graph Neural Network with Supervised Contrastive Regularization](https://mlsanity.com/paper/2404.06090/) (2024)
- [Privilege Scores](https://mlsanity.com/paper/2502.01211/) (2025)
- [DISCOVER: A Solver for Distributional Counterfactual Explanations](https://mlsanity.com/paper/2603.16436/) (2026)
- [Fairness constraint in Structural Econometrics and Application to fair estimation using Instrumental Variables](https://mlsanity.com/paper/2202.08977/) (2022)
- [Local and Regional Counterfactual Rules: Summarized and Robust Recourses](https://mlsanity.com/paper/2209.14568/) (2022)
- [Optimal Transport on Categorical Data for Counterfactuals using Compositional Data and Dirichlet Transport](https://mlsanity.com/paper/2501.15549/) (2025)
- [Learning Counterfactually Fair Models via Improved Generation with Neural Causal Models](https://mlsanity.com/paper/2502.12796/) (2025)
- [PLS-based approach for fair representation learning](https://mlsanity.com/paper/2502.16263/) (2025)
- [What is a good matching of probability measures? A counterfactual lens on transport maps](https://mlsanity.com/paper/2509.16027/) (2025)
- [A pipeline for enabling path-specific causal fairness in observational health data](https://mlsanity.com/paper/2601.09841/) (2026)
- [Optimal Transport Group Counterfactual Explanations](https://mlsanity.com/paper/2601.20692/) (2026)
- [Causally Sufficient and Necessary Feature Expansion for Class-Incremental Learning](https://mlsanity.com/paper/2603.09145/) (2026)
- [Homophily-aware Supervised Contrastive Counterfactual Augmented Fair Graph Neural Network](https://mlsanity.com/paper/2604.02342/) (2026)

## Abstract (arXiv)

Counterfactual frameworks have grown popular in machine learning for both explaining algorithmic decisions but also defining individual notions of fairness, more intuitive than typical group fairness conditions. However, state-of-the-art models to compute counterfactuals are either unrealistic or unfeasible. In particular, while Pearl's causal inference provides appealing rules to calculate counterfactuals, it relies on a model that is unknown and hard to discover in practice. We address the problem of designing realistic and feasible counterfactuals in the absence of a causal model. We define transport-based counterfactual models as collections of joint probability distributions between observable distributions, and show their connection to causal counterfactuals. More specifically, we argue that optimal-transport theory defines relevant transport-based counterfactual models, as they are numerically feasible, statistically-faithful, and can coincide under some assumptions with causal counterfactual models. Finally, these models make counterfactual approaches to fairness feasible, and we illustrate their practicality and efficiency on fair learning. With this paper, we aim at laying out the theoretical foundations for a new, implementable approach to counterfactual thinking.