# A primer on optimal transport for causal inference with observational data

arXiv: https://arxiv.org/abs/2503.07811 · HTML: https://mlsanity.com/paper/2503.07811/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 152 references that are in the map.
- First posted on arXiv 10 Mar 2025.

## Builds on (most cited first, 28 of 28)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Counterfactual Probabilities: Computational Methods, Bounds and Applications](https://mlsanity.com/paper/1302.6784/) (2013)
- [Unbalanced Optimal Transport, from Theory to Numerics](https://mlsanity.com/paper/2211.08775/) (2022)
- [Synthetic Difference in Differences](https://mlsanity.com/paper/1812.09970/) (2018)
- [Generalized Optimal Matching Methods for Causal Inference](https://mlsanity.com/paper/1612.08321/) (2016)
- [A Class of Algorithms for General Instrumental Variable Models](https://mlsanity.com/paper/2006.06366/) (2020)
- [Transport-based Counterfactual Models](https://mlsanity.com/paper/2108.13025/) (2021)
- [On the Testability of Causal Models with Latent and Instrumental Variables](https://mlsanity.com/paper/1302.4976/) (2013)
- [Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space](https://mlsanity.com/paper/1910.05954/) (2019)
- [The Balancing Act in Causal Inference](https://mlsanity.com/paper/2110.14831/) (2021)
- [Optimal transport for causal discovery](https://mlsanity.com/paper/2201.09366/) (2022)
- [What's Trending in Difference-in-Differences? A Synthesis of the Recent Econometrics Literature](https://mlsanity.com/paper/2201.01194/) (2022)
- [Structural Nested Models and G-estimation: The Partially Realized Promise](https://mlsanity.com/paper/1503.01589/) (2015)
- [Causal Inference on Distribution Functions](https://mlsanity.com/paper/2101.01599/) (2021)
- [Nonparametric Multiple-Output Center-Outward Quantile Regression](https://mlsanity.com/paper/2204.11756/) (2022)
- [Distributional synthetic controls](https://mlsanity.com/paper/2001.06118/) (2020)
- [Measure Estimation in the Barycentric Coding Model](https://mlsanity.com/paper/2201.12195/) (2022)
- [Optimal transport and Wasserstein distances for causal models](https://mlsanity.com/paper/2303.14085/) (2023)
- [Adaptive Hyper-box Matching for Interpretable Individualized Treatment Effect Estimation](https://mlsanity.com/paper/2003.01805/) (2020)
- [An Optimal Transport Approach to Estimating Causal Effects via Nonlinear Difference-in-Differences](https://mlsanity.com/paper/2108.05858/) (2021)
- [Optimal transportation between unequal dimensions](https://mlsanity.com/paper/1805.11187/) (2018)
- [When Is Parallel Trends Sensitive to Functional Form?](https://mlsanity.com/paper/2010.04814/) (2020)
- [Generating Synthetic Datasets by Interpolating along Generalized Geodesics](https://mlsanity.com/paper/2306.06866/) (2023)
- [Matching for causal effects via multimarginal unbalanced optimal transport](https://mlsanity.com/paper/2112.04398/) (2021)
- [Conservative Inference for Counterfactuals](https://mlsanity.com/paper/2310.12757/) (2023)
- [Lorenz map, inequality ordering and curves based on multidimensional rearrangements](https://mlsanity.com/paper/2203.09000/) (2022)

## Built on by (most cited first, 2 of 2)

- [Wasserstein Parallel Transport for Predicting the Dynamics of Statistical Systems](https://mlsanity.com/paper/2603.23736/) (2026)
- [Partial identification with entropy regularized optimal transport](https://mlsanity.com/paper/2609.40156/) (2026)

## Abstract (arXiv)

The theory of optimal transportation has developed into a powerful and elegant framework for comparing probability distributions, with wide-ranging applications in all areas of science. The fundamental idea of analyzing probabilities by comparing their underlying state space naturally aligns with the core idea of causal inference, where understanding and quantifying counterfactual states is paramount. Despite this intuitive connection, explicit research at the intersection of optimal transport and causal inference is only beginning to develop. Yet, many foundational models in causal inference have implicitly relied on optimal transport principles for decades, without recognizing the underlying connection. Therefore, the goal of this review is to offer an introduction to the surprisingly deep existing connections between optimal transport and the identification of causal effects with observational data -- where optimal transport is not just a set of potential tools, but actually builds the foundation of model assumptions. As a result, this review is intended to unify the language and notation between different areas of statistics, mathematics, and econometrics, by pointing out these existing connections, and to explore novel problems and directions for future work in both areas derived from this realization.