Dictator Functions Maximize Mutual Information

Georg Pichler; Pablo Piantanida; Gerald Matz

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arXiv · 1604.02109

Dictator Functions Maximize Mutual Information

Georg Pichler·Pablo Piantanida·Gerald Matz

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Abstract

Let $(\mathbf X, \mathbf Y)$ denote $n$ independent, identically distributed copies of two arbitrarily correlated Rademacher random variables $(X, Y)$. We prove that the inequality $I(f(\mathbf X); g(\mathbf Y)) \le I(X; Y)$ holds for any two Boolean functions: $f,g \colon \{-1,1\}^n \to \{-1,1\}$ ($I(\cdot; \cdot)$ denotes mutual information). We further show that equality in general is achieved only by the dictator functions $f(\mathbf x)=\pm g(\mathbf x)=\pm x_i$, $i \in \{1,2,\dots,n\}$.

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Georg Pichler, Pablo Piantanida, Gerald Matz. 2016-04-07. Dictator Functions Maximize Mutual Information. https://arxiv.org/abs/1604.02109

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