Bridging Multiple Worlds: Multi-marginal Optimal Transport for Causal Partial-identification Problem

Under the prevalent potential outcome model in causal inference, each unit is associated with multiple potential outcomes but at most one of which is observed, leading to many causal quantities being only partially identified. The inherent missing data issue echoes the multi-marginal optimal transport (MOT) problem, where marginal distributions are known, but how the marginals couple to form the joint distribution is unavailable. In this paper, we cast the causal partial identification problem in the framework of MOT with $K$ margins and $d$-dimensional outcomes and obtain the exact partial identified set. In order to estimate the partial identified set via MOT, statistically,we establish a convergence rate of the plug-in MOT estimator for the $\ell_2$ cost function stemming from the variance minimization problem and prove it is minimax optimal for arbitrary $K$ and $d \le 4$. We also extend the convergence result to general quadratic objective functions. Numerically, we demonstrate the efficacy of our method over synthetic datasets and several real-world datasets where our proposal consistently outperforms the baseline by a significant margin (over 70\%). In addition, we provide efficient off-the-shelf implementations of MOT with general objective functions.