Mrs. Gerbers Lemma

Mrs. Gerber's Lemma is a result about the entropy of binary sequences.

A convexity lemma
Let $h_b(\cdot)$ denote the binary entropy function, $u \in [0,\log 2]$, and $p_0 \in (0,1/2)$.

Lemma. The function $h_b(p_0 \ast h_b^{-1}(u))$ is strictly convex in $u$, where \begin{align} a \ast b = a (1 - b) + (1 -a) b. \end{align}

The main lemma
Let $X^n = (X_1, X_2, \ldots, X_n)$ be a binary random vector with distribution $P(X^n)$ such that the entropy \begin{align} H(X^n) = - \sum_{x^n} P(x^n) \log P(x^n) \ge n v \end{align} for some $v \in [0,\log2]$. Suppose $X^n$ is transmitted through a binary symmetric channel with crossover probability $p_0$ and let $Y^n$ denote the output of this channel. That is, $Y^n = X^n \oplus Z^n$ where $Z^n$ is an i.i.d. binary vector with $\mathbb{P}(Z_i = 1) = p_0$ and $\oplus$ denotes modulo-two addition.

Lemma. For a binary random vector $X^n$ such that $H(X^n) \ge v n$ and random variable $Y^n = X^n \oplus Z^n$ where $Z^n$ is i.i.d. with $\mathbb{P}(Z_i = 1) = p_0$, \begin{align} H(Y^n) \ge n h_b( p_0 \ast h^{-1}(v) ) \end{align} with equality if and only if $X^n$ is i.i.d. with $\mathbb{P}(X_i = 1) = v$.

History
Mrs. Gerber's Lemma was introduced by Wyner and Ziv in 1973. A footnote to the main result (Theorem 1) of that paper stated: "This result is known as “Mrs. Gerber’s Lemma” in honor of a certain lady whose presence was keenly felt by the authors at the time this research was done." It was later generalized by Shamai and Wyner as a binary version of the entropy power inequality.