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I'm trying to make sense of the Wasserstein metric, it says in wikipedia

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Say the marginal distributions $\mu$ and $\nu$ are given, what properties does a joint distribution from the two need to satisfy? For example like $\mu(x)=\int p(x,y)dx$ and $\nu(y)=\int p(x,y)dy$?

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The joint distribution just needs to be a valid probability distribution, with matching marginals. That's it.

It turns out that you can characterize the set of such distributions via copulas. That is, if we assume for now that $X$ and $Y$ are each real-valued, every such distribution has cdf of the form $$ \Pr(X \le x, Y \le y) = C(\Pr(X \le x), \Pr(Y \le y)) $$ where $C$ is the joint cdf of a distribution that has uniform margins on the unit square. The converse also holds: any such $C$ gives you a valid joint distribution. I'm pretty sure something analogous holds for $X$ and $Y$ taking values in $\mathbb R^n$, but apparently it only holds in one direction for infinite-dimensional Hilbert spaces.

That doesn't necessarily help you in computing the Wasserstein distance, but it might help in thinking about what it means.

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    $\begingroup$ Ohhh, cool paper! $\endgroup$ Commented Feb 13, 2017 at 15:14

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