Dictator functions maximize mutual information

Abstract : Let (Xn,Yn) denote n independent, identically distributed copies of two arbitrarily correlated Rademacher random variables (X,Y) on {−1,1}. We prove that the inequality I(f(Xn);g(Yn))≤I(X;Y) holds for any two Boolean functions: f,g:{−1,1}n→{−1,1} (I(⋅;⋅) denotes mutual information.) We further show that equality in general is achieved only by the dictator functions: f(x)=±g(x)=±xi for every i∈{1,2,…,n}.
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The Annals of Applied Probability : an official journal of the institute of mathematical statistics, The Institute of Mathematical Statistics, 2018, 28 (5), pp.3094-3101. 〈10.1214/18-aap1384 〉
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Contributeur : Pablo Piantanida <>
Soumis le : lundi 16 janvier 2017 - 16:41:33
Dernière modification le : jeudi 22 novembre 2018 - 13:16:01

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Pichler Georg, Pablo Piantanida, Gerald Matz. Dictator functions maximize mutual information. The Annals of Applied Probability : an official journal of the institute of mathematical statistics, The Institute of Mathematical Statistics, 2018, 28 (5), pp.3094-3101. 〈10.1214/18-aap1384 〉. 〈hal-01436725〉

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