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 by the average entropy of conditional expectations of f , given sets of roughly (1 − 2 ) 2 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 , then, provided 1 2 − for a sufficiently small positive constant , for any balanced boolean function f holds I f ( X ); Y 1 − H ( ) .