The Annals of Applied Probability

Dictator functions maximize mutual information

Georg Pichler, Pablo Piantanida, and Gerald Matz

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

Article information

Ann. Appl. Probab., Volume 28, Number 5 (2018), 3094-3101.

Received: September 2016
Revised: January 2018
First available in Project Euclid: 28 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 94A15: Information theory, general [See also 62B10, 81P94]
Secondary: 94C10: Switching theory, application of Boolean algebra; Boolean functions [See also 06E30]

Boolean functions mutual information Fourier analysis binary sequences binary codes


Pichler, Georg; Piantanida, Pablo; Matz, Gerald. Dictator functions maximize mutual information. Ann. Appl. Probab. 28 (2018), no. 5, 3094--3101. doi:10.1214/18-AAP1384.

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