Approximation with independent Variables

Abstract

Given a square integrable m-dimensional random variable $ X $ on a probability space $ (\Omega, {{\mathcal{F}}}, {{\mathbb P}}) $ 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 $. The construction of $ Y $ is done in two steps. First, using transport theory we determine the distribution of $ Y $, in a second step we construct the variable $ Y $ using the existence of a uniformly $ [0, 1] $ distributed random variable independent of $ {{\mathcal{A}}} $. Such results have an importance to fairness and bias reduction in Artificial Intelligence, Machine Learning and Network Theory. We believe that the problem was never studied in the past.