# Controlling Mean Difference Between Product and Joint Distributions Using Optimal Transportation

Suppose we have nonindependent random variables $$X \sim P$$ and $$Y \sim Q$$, where $$P$$ and $$Q$$ denote their marginal distributions. We are interested in upper bounding $$|\mathbf{E}_{X, Y\sim P \otimes Q } f(X, Y) - \mathbf{E}_{X, Y\sim P, Q } f(X, Y)|$$ for a given function $$f$$. Here $$P \otimes Q$$ denotes the product distribution of $$P$$ and $$Q$$, while $$P, Q$$ denotes their joint distribution (or equivalently a coupling).

Question: Can we upper bound such a mean difference using the Wasserstein distance between $$P$$ and $$Q$$, or the Wasserstein distance between $$P \otimes Q$$ and $$P, Q$$?

If needed, we are allowed to impose regularity conditions, such as Lipschitz continuity, on $$f$$.

By the dual representation of the Wasserstein metric $$W_1$$, $$|\mathbf{E}_{X, Y\sim P \otimes Q } f(X, Y) - \mathbf{E}_{X, Y\sim P, Q } f(X, Y)|\le\text{Lip}(f)\,W_1(P \otimes Q,(P, Q)),$$ where $$\text{Lip}(f)$$ is the Lipschitz constant of $$f$$.
On the other hand, the closeness of $$P$$ to $$Q$$ in any metric will not by itself provide any interesting upper bound on $$|\mathbf{E}_{X, Y\sim P \otimes Q } f(X, Y) - \mathbf{E}_{X, Y\sim P, Q } f(X, Y)|$$; e.g., take $$P=Q$$.