How to quantize n outputs of a binary symmetric channel to n - 1 bits?
نویسندگان
چکیده
Suppose that Y n is obtained by observing a uniform Bernoulli random vector X through a binary symmetric channel with crossover probability α. The “most informative Boolean function” conjecture postulates that the maximal mutual information between Y n and any Boolean function b(X) 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} → {0, 1}, namely, an n − 1 bit quantizer, and show that I(f(X);Y ) ≤ (n − 1) · (1− h(α)) for any such f . Thus, in this case, the optimal function is of the form f(x) = (x1, . . . , xn−1).
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