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October 2018 Dictator functions maximize mutual information
Georg Pichler, Pablo Piantanida, Gerald Matz
Ann. Appl. Probab. 28(5): 3094-3101 (October 2018). DOI: 10.1214/18-AAP1384

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 $f(\boldsymbol{x})=\pm g(\boldsymbol{x})=\pm x_{i}$, $i\in \{1,2,\ldots,n\}$.

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Georg Pichler. Pablo Piantanida. Gerald Matz. "Dictator functions maximize mutual information." Ann. Appl. Probab. 28 (5) 3094 - 3101, October 2018. https://doi.org/10.1214/18-AAP1384

Information

Received: 1 September 2016; Revised: 1 January 2018; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974774
MathSciNet: MR3847982
Digital Object Identifier: 10.1214/18-AAP1384

Subjects:
Primary: 94A15
Secondary: 94C10

Keywords: binary codes , binary sequences , Boolean functions , Fourier analysis , mutual information

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 5 • October 2018
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