Approximation with Independent Variables

Abstract

Given a square integrable m-dimensional random variable $X$ on a probability space $(\Omega .\mathcal{F},\mathrm{Pr})$ 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.