A local-to-global principle for $f$-divergence homogenization

Abstract

We study the effect of replacing a family of independent statistical experiments by independent repetitions of their average. We give a simple, complete characterization of the $f$-divergences for which this operation increases divergence by at most a constant depending only on the generator, uniformly in the number of observations and the underlying measurable space. The characterization is governed by a local-to-global principle: a comparison restricted to two observations on a binary alphabet already determines the validity of the general inequality. It admits an explicit scalar formulation, which simplifies further for symmetric divergences. The result applies to arbitrary convex generators and requires no differentiability or symmetry assumptions. It recovers the homogenization inequality for total variation and identifies exactly the admissible members of the power-divergence family. In particular, whenever uniform homogenization fails, the obstruction is already present in a family of binary experiments with two observations. Moreover, the global comparison constant is controlled uniformly over generators with a common bound in the local inequality.