# Couplings on empirical distributions

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?

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.
• 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. – Glassjawed Mar 5 at 17:09
• "$\pi$ 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 $\hat\mu_T$? – Iosif Pinelis Mar 5 at 17:17
• 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. – Glassjawed Mar 5 at 18:05