Approximation with Independent Variables

TL;DR

This paper proves the existence of an independent approximation of a given random variable that minimizes the L^2 distance, with implications for fairness and bias reduction in AI and related fields.

Contribution

It introduces a method to construct an independent random variable closest to a given one in L^2 sense, advancing understanding of independence approximation.

Findings

Existence of an independent approximation minimizing L^2 distance.

Relevance to fairness and bias reduction in AI and machine learning.

Potential applications in network theory and related areas.

Abstract

Given a square integrable m-dimensional random variable $X$ on a probability space $(\Omega.\mathcal F,\Pr)$ and a sub sigma algebra $\mathcal A$, we show that there exists another m-dimensional random variable $Y$, independent of $\mathcal A$ and minimising the $L^2$ distance to $X$. Such results have an importance to fairness and bias reduction in Artificial Intelligence, Machine Learning and Network Theory.