## Abstract

Let 0 < ϵ < 1/2 be a noise parameter, and let T_{ϵ} 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_{ϵ} f by the average entropy of conditional expectations of f, given sets of roughly (1-2 ϵ )^{2}. 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 ϵ . Then, setting v = (1-2 ϵ)^{2}. n, we have (up to lower order terms): H (X ⊕ Z) ≥ n H_{2} ( ϵ + (1-2 ϵ) H_{2}^{-1} (E_{|B|} = v H (X_{i}{i ∈ B)/v). Assuming ϵ ≥ 1/2-δ, for some absolute constant δ > 0, this inequality, combined with a strong version of a theorem of Friedgut et al., due to Jendrej et al., shows that if a Boolean function f is close to a characteristic function g of a subcube of dimension n-1, then the entropy of T_{ϵ} f is at most that of T_{ϵ} g. Taken together with a recent result of Ordentlich et al., this shows that the most informative Boolean function conjecture of Courtade and Kumar holds for high noise ϵ ≥ 1/2-δ. Namely, if X is uniformly distributed in {0,1^{n}} and Y is obtained by flipping each coordinate of X independently with probability ϵ, then, provided ϵ ≥ 1/2-δ, for any Boolean function f holds I (f(X);Y ) ≤ 1-H(ϵ).

Original language | English |
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Article number | 7498615 |

Pages (from-to) | 5446-5464 |

Number of pages | 19 |

Journal | IEEE Transactions on Information Theory |

Volume | 62 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2016 |

### Bibliographical note

Publisher Copyright:© 2016 IEEE.

## Keywords

- Boolean functions
- extremal inequality
- mutual information