Suppose that Yn is obtained by observing a uniform Bernoulli random vector
Xn through a binary symmetric channel with crossover probability Ξ±.
The "most informative Boolean function" conjecture postulates that the maximal
mutual information between Yn and any Boolean function b(Xn) is
attained by a dictator function. In this paper, we consider the "complementary"
case in which the Boolean function is replaced by
f:{0,1}nβ{0,1}nβ1, namely, an nβ1 bit
quantizer, and show that I(f(Xn);Yn)β€(nβ1)β (1βh(Ξ±))
for any such f. Thus, in this case, the optimal function is of the form
f(xn)=(x1β,β¦,xnβ1β).Comment: 5 pages, accepted ISIT 201