Optimal use of auxiliary information : information geometry and empirical process

TL;DR

This paper introduces an optimal information geometry framework to incorporate auxiliary expectation information into empirical measures, unifying various methods and demonstrating asymptotic properties and variance reduction.

Contribution

It develops a unified approach to integrate auxiliary information into empirical measures using information geometry, establishing asymptotic theorems and demonstrating variance reduction.

Findings

Established Glivenko-Cantelli and Donsker theorems for informed empirical measures.

Proved the informed empirical process is more concentrated than the classical one.

Quantified asymptotic variance reduction and applied results to informed empirical quantiles.

Abstract

We incorporate into the empirical measure the auxiliary information given by a finite collection of expectation in an optimal information geometry way. This allows to unify several methods exploiting a side information and to uniquely define an informed empirical measure. These methods are shown to share the same asymptotic properties. Then we study the informed empirical process subject to a true information. We establish the Glivenko-Cantelli and Donsker theorems for the informed empirical measure under minimal assumptions and we quantify the asymptotic uniform variance reduction. Moreover, we prove that the informed empirical process is more concentrated than the classical empirical process for all large $n$. Finally, as an illustration of the variance reduction, we apply some of these results to the informed empirical quantiles.