Which Boolean Functions are Most Informative?

TL;DR

This paper explores a conjecture about the maximum mutual information that Boolean functions can reveal about noisy inputs, specifically in the context of binary symmetric channels, and provides evidence supporting its validity.

Contribution

It introduces a conjecture on the upper bound of mutual information for Boolean functions under noise and offers substantial evidence for its correctness.

Findings

Conjecture that mutual information is bounded by 1-H(α) for Boolean functions.

Supporting evidence suggests the conjecture may be true.

The problem remains open for a formal proof.

Abstract

We introduce a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let $X^n$ be i.i.d. Bernoulli(1/2), and let $Y^n$ be the result of passing $X^n$ through a memoryless binary symmetric channel with crossover probability $\alpha$. For any Boolean function $b:\{0,1\}^n\rightarrow \{0,1\}$, we conjecture that $I(b(X^n);Y^n)\leq 1-H(\alpha)$. While the conjecture remains open, we provide substantial evidence supporting its validity.