Fundamental limits of the sensitivity of unbiased estimation through the lens of the Wasserstein geometry

Music City Center 

In this talk, we revisit the classical theory of variance bounds for unbiased estimators and develop an analogous framework for a different notion of estimator stability, which we term sensitivity. Roughly speaking, the sensitivity of an estimator quantifies how much its output changes when its inputs are perturbed slightly. In the same way that classical inequalities like the Cramer-Rao lower bound for the variance of unbiased estimators can be interpreted geometrically via the Fisher-Rao geometry, our notion of sensitivity admits analogous inequalities and theory, only that now they arise from Wasserstein geometry. We will discuss this Wasserstein-Cramer-Rao lower bound and its associated notion of Wasserstein Fisher information (which are closely related to notions introduced by Li and Zhao, 2023), introduce the notion of Wasserstein exponential families and their defining properties, and introduce the concept of Wasserstein MLEs. In particular, we will discuss that the Wasserstein MLE is, generically, asymptotically efficient for sensitivity, in the sense that it achieves the corresponding Wasserstein-Cramer-Rao lower bound. Our theory reveals new optimality properties for existing estimators and, in other cases, reveals entirely new estimators.

This is joint work with Adam Quinn Jaffe and Bodhisattva Sen.

optimal transport

data analysis

vectorial optimal transport

linearization