Let $f$ be a nonnegative function on $\{0,1\}^n$. We upper bound the entropy of the image of $f$ under the noise operator with noise parameter $\epsilon$ by the average entropy of conditional expectations of $f$, given sets of roughly $(1-2\epsilon)^2 \cdot n$ variables.
As an application, we show that for a boolean function $f$, which is close to a characteristic function $g$ of a subcube of dimension $n-1$, the entropy of the noisy version of $f$ is at most that of the noisy version of $g$.
This, combined with a recent result of Ordentlich, Shayevitz, and Weinstein, shows that the "Most informative boolean function" conjecture of Courtade and Kumar holds for high noise (close to $1/2$).
Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $\epsilon$, then, provided $\epsilon \ge 1/2 - \delta$ for a sufficiently small positive constant $\delta$, for any boolean function $f$ holds $I\Big(f(X);Y\Big) \le 1 - H(\epsilon)$.
Significantly revised and merged with a follow-up paper.
Let $f$ be a nonnegative function on $\{0,1\}^n$. We upper bound the entropy of the image of $f$ under the noise operator with noise parameter $\epsilon$ by the average entropy of conditional expectations of $f$, given sets of roughly $(1-2\epsilon)^2 \cdot n$ variables.
As an application, we show that for a boolean function $f$, which is close to a characteristic function $g$ of a subcube of dimension $n-1$, the entropy of the noisy version of $f$ is at most that of the noisy version of $g$.
This, combined with a recent result of Ordentlich, Shayevitz, and Weinstein, shows that the "Most informative boolean function" conjecture of Courtade and Kumar holds for balanced boolean functions and high noise (close to $1/2$).
Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $\epsilon$, then, provided $\epsilon \ge 1/2 - \delta$ for a sufficiently small positive constant $\delta$, for any balanced boolean function $f$ holds $I\Big(f(X);Y\Big) \le 1 - H(\epsilon)$.
