TV homogenization inequalities

TL;DR

This paper investigates how homogenization affects total variation distance in Bernoulli measures, showing it reduces TV distance up to a universal constant despite not being a Markov kernel.

Contribution

It proves that homogenization, unlike summation, still decreases TV distance with a universal bound, using explicit control of Poisson binomial TV distances.

Findings

Homogenization reduces total variation distance up to a universal constant.

Explicit bounds on TV distance between Poisson binomials are established.

Homogenization does not satisfy the data processing inequality but still contracts TV distance.

Abstract

We study the total variation distance under two information-erasing maps on inhomogeneous Bernoulli product measures: summation and homogenization. While summation is a Markov kernel and hence satisfies the usual data processing inequality, homogenization -- which maps each Bernoulli parameter to the cumulative mean -- is not. Nevertheless, we prove that the homogenization map also reduces the TV distance, up to a universal constant. The argument is based on an explicit two-sided control of the TV distance between Poisson binomials, obtained via a parameter interpolation and a second-moment extraction lemma.