Dictator functions maximize mutual information

Abstract

Let $(\boldsymbol{\mathsf{X}},\boldsymbol{\mathsf{Y}})$ denote $n$ independent, identically distributed copies of two arbitrarily correlated Rademacher random variables $(\mathsf{X},\mathsf{Y})$. We prove that the inequality $\mathrm{I}(f(\boldsymbol{\mathsf{X}});g(\boldsymbol{\mathsf{Y}}))\le \mathrm{I}(\mathsf{X};\mathsf{Y})$ holds for any two Boolean functions: $f,g\colon \{-1,1\}^{n}\to \{-1,1\}$ [$\mathrm{I}(\cdot ;\cdot)$ denotes mutual information]. We further show that equality in general is achieved only by the dictator functions $f(\boldsymbol{x})=\pm g(\boldsymbol{x})=\pm x_{i}$, $i\in \{1,2,\ldots,n\}$.