TV homogenization inequalities
Aryeh Kontorovich
TL;DR
The paper analyzes how total variation distance between inhomogeneous Bernoulli product measures behaves under two information-erasing maps: summation and homogenization. It introduces the Poisson-binomial framework with key quantities Delta, sigma^2 and Phi to quantify TV between S_p and S_q, and proves sharp, universal-bounded relations: an upper bound and a matching lower bound for dominating pairs, establishing that TV is controlled by Phi up to universal constants. Building on this, it proves a homogenization inequality that lower-bounds TV between Bernoulli product measures by a constant multiple of the TV between the corresponding binomial distributions with mean parameters p_bar and q_bar, using a binomial homogenization lemma and a heavy-atom/pigeonhole argument. The results provide a quantitative link between information-erasure via summation and via mean-constraining homogenization, with explicit constants (C_BCV and c) and a conjectured optimal homogenization constant 8/9, highlighting new structural insights into Poisson-binomial sums and their role in information-theoretic distance measures.
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.