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In 2013, Courtade and Kumar posed the following problem: Let $\boldsymbol{x}\sim \{\pm 1\}^n$ be uniformly random, and form $\boldsymbol{y} \sim \{\pm1\}^n$ by negating each bit of $\boldsymbol{x}$ independently with probability$\alpha$. Is it true that the mutual information $I(f(\boldsymbol{x})\mathbin{;} \boldsymbol{y})$ is maximized among $f:\{\pm 1\}^n \to \{\pm 1\}$by $f(x) = x_1$? We do not resolve this problem. Instead, we make a couple ofobservations about the fixed-mean version of the conjecture. We show thatCourtade and Kumar's stronger Lex Conjecture fails for small noise rates. Wealso prove a continuous version of the conjecture on the sphere and show thatit implies the previously-known analogue for Gaussian space.

Remarks on the Most Informative Function Conjecture at fixed mean