Wasserstein information matrix

TL;DR

This paper introduces Wasserstein information matrices (WIMs) as a new way to analyze statistical models using the $L^2$-Wasserstein metric, establishing bounds and efficiency results for estimators.

Contribution

It develops the theory of WIMs, including score functions, covariance operators, and bounds, and applies these concepts to various models and estimators.

Findings

WIMs provide bounds analogous to Fisher information bounds.

Wasserstein natural gradient achieves on-line asymptotic efficiency.

Analytical examples demonstrate WIMs in different model families.

Abstract

We study information matrices for statistical models by the $L^2$-Wasserstein metric. We call them Wasserstein information matrices (WIMs), which are analogs of classical Fisher information matrices. We introduce Wasserstein score functions and study covariance operators in statistical models. Using them, we establish Wasserstein-Cramer-Rao bounds for estimations and explore their comparisons with classical results. We next consider the asymptotic behaviors and efficiency of estimators. We derive the on-line asymptotic efficiency for Wasserstein natural gradient. Besides, we study a Poincar\'e efficiency for Wasserstein natural gradient of maximal likelihood estimation. Several analytical examples of WIMs are presented, including location-scale families, independent families, and rectified linear unit (ReLU) generative models.