  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 general quadratic
objective functions and prove it is minimax optimal for a quadratic objective
function stemming from the variance minimization problem with arbitrary $K$ and
$d \le 4$. Numerically, we demonstrate the efficacy of our method over 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.
