On the entropy of a noisy function

Let $0 < \epsilon < 1/2$ be a noise parameter, and let $T_{\epsilon}$ be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a nonnegative function on $\{0,1\}^n$. We upper bound the entropy of $T_{\epsilon} f$ by the average entropy of conditional expectations of $f$, given sets of roughly $(1-2\epsilon)^2 \cdot n$ variables. In information-theoretic terms, we prove the following strengthening of "Mrs. Gerber's lemma": Let $X$ be a random binary vector of length $n$, and let $Z$ be a noise vector, corresponding to a binary symmetric channel with crossover probability $\epsilon$. Then, setting $v = (1-2\epsilon)^2 \cdot n$, we have (up to lower-order terms): $$H\Big(X \oplus Z\Big) \ge n \cdot H\left(\epsilon ~+~ (1-2\epsilon) \cdot H^{-1}\left(\frac{{\mathbb E}_{|B| = v} H\Big(\{X_i\}_{i\in B}\Big)}{v}\right)\right)$$ 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 $T_{\epsilon} f$ is at most that of $T_{\epsilon} g$. 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 $\epsilon \ge 1/2 - \delta$, for some absolute constant $\delta > 0$. 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 any boolean function $f$ holds $I\Big(f(X);Y\Big) \le 1 - H(\epsilon)$.