How do we do optimal transport when we don't know the data?

Optimal transport is powerful for comparing probability distributions, but real applications work with noisy, finite samples—not perfect data. This work lifts optimal transport theory to handle random probability measures, introducing a new space (L² over Wasserstein) that preserves the geometric structure while letting uncertainty flow through. The framework unifies sampling-based inference and connects to recent transformer attention mechanisms, showing how uncertainty propagates in both Bayesian non-parametrics and generative models.