  We study multi-marginal optimal transport problems from a probabilistic
graphical model perspective. We point out an elegant connection between the two
when the underlying cost for optimal transport allows a graph structure. In
particular, an entropy regularized multi-marginal optimal transport is
equivalent to a Bayesian marginal inference problem for probabilistic graphical
models with the additional requirement that some of the marginal distributions
are specified. This relation on the one hand extends the optimal transport as
well as the probabilistic graphical model theories, and on the other hand leads
to fast algorithms for multi-marginal optimal transport by leveraging the
well-developed algorithms in Bayesian inference. Several numerical examples are
provided to highlight the results.
