A primer on optimal transport for causal inference with observational data
This work addresses the need for a unified framework in statistics, mathematics, and econometrics by clarifying existing connections, but it is incremental as it reviews and synthesizes prior knowledge rather than introducing new methods.
This review tackles the problem of connecting optimal transport theory with causal inference for observational data, revealing that foundational causal models have implicitly relied on optimal transport principles for decades without explicit recognition, and it aims to unify notation and explore future directions.
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.