Dictator functions maximize mutual information
Résumé
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}.
Origine : Fichiers produits par l'(les) auteur(s)
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