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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Contributor : Pablo Piantanida <>
Submitted on : Monday, January 16, 2017 - 4:41:33 PM
Last modification on : Monday, February 18, 2019 - 7:52:11 PM

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Pichler Georg, Pablo Piantanida, Gerald Matz. Dictator functions maximize mutual information. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2018, 28 (5), pp.3094-3101. ⟨10.1214/18-aap1384 ⟩. ⟨hal-01436725⟩

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