Optimal transport (OT) is a useful tool for describing the geometry of the space of probability measures. This talk presents two applications of OT-based methods to nonparametric hypothesis testing. First, we develop a theory of simultaneous OT, which resolves a long-standing problem on the existence of p-values for composite hypothesis testing and provides an explicit algorithm for constructing powerful p-values when they exist. Second, using OT projection under causality constraints, we develop the first consistent test of the martingale pair property from iid samples. The test can be applied in various settings, including non-arbitrage tests in generative models of prices in financial markets. An important feature is that, while the martingale property involves infinitely many constraints, our test avoids the curse of dimensionality. 

This is based on joint works with Jose Blanchet, Aaditya Ramdas, Ruodu Wang, Johannes Wiesel, and Erica Zhang.