Dictator Functions Maximize Mutual Information

TL;DR

This paper proves that for correlated Rademacher variables, the mutual information between Boolean functions is maximized only by dictator functions, establishing a fundamental inequality and characterizing equality cases.

Contribution

The paper establishes a mutual information inequality for Boolean functions of correlated variables and characterizes the conditions for equality, identifying dictator functions as optimal.

Findings

Mutual information between Boolean functions is bounded by the original variables' mutual information.

Equality in the inequality is achieved only by dictator functions.

The result applies to arbitrarily correlated Rademacher variables.

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 $f(\mathbf x)=\pm g(\mathbf x)=\pm x_i$, $i \in \{1,2,\dots,n\}$.