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For a problem I've been working on, I'm thinking about couplings between true and empirical distrubutions.

I have two datasets $S$ and $T$ with underlying measures $\mu_S,\,\mu_T$. And then I have some coupling on those, say $\pi\in \Pi(\mu_S,\mu_T)$ with $\mu_S$ and $\mu_T$ as marginals.

Now if I come up with empirical distributions $\hat{\mu}_S$ and $\hat{\mu}_T$ based off of sample draws from $S$, $T$, is there anything I can say about the couplings $\hat{\pi} \in \Pi(\hat{\mu_S},\hat{\mu_T})$?

Or at least any literature out there that discusses this?

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If $\pi$ is known and you want $\hat\pi$ to be close to $\pi$, then you can define $\hat\pi$ as a minimizer of a distance $d(\hat\pi,\pi)$ from $\hat\pi$ to $\pi$ over all $\hat\pi\in\Pi(\hat{\mu_S},\hat{\mu_T})$. For $d$ you can use e.g. a Wasserstein distance.

On the other hand, if $\pi$ is known (and hence $\mu_S$ and $\mu_T$ are known), why would you need empirical measures at all? Empirical measures are used to estimate their true, but unknown counterparts.

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  • $\begingroup$ My bad, I meant to say $\pi$ is the Wasserstein coupling—i.e. $W(S,T)=\int c(x_s,x_t)\, d\pi(x_s,x_t)$ where $c$ is some “cost” function. $\endgroup$ – Glassjawed Mar 5 at 17:09
  • $\begingroup$ "$\pi$ is the Wasserstein coupling"... between what and what? If between the measures $\mu_S$ and $\mu_T$, and you know these measures, why would you then need $\hat\mu_S$ and $\hat\mu_T$? $\endgroup$ – Iosif Pinelis Mar 5 at 17:17
  • $\begingroup$ It’s for a research problem I have where I’m trying to understand empirical losses. $\pi$ is the Wasserstein coupling between $\mu_S$ and $\mu_T$. We don’t necessarily know the those measures but we can use concentration bounds between empirical and true losses. $\endgroup$ – Glassjawed Mar 5 at 18:05

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