Guessing with a Bit of Help

TL;DR

This paper investigates how much a single bit of information can improve guessing a random vector observed through a noisy channel, providing bounds and conjectures on optimal guessing strategies.

Contribution

It introduces bounds on guessing efficiency for binary vectors observed through channels, extending to general channels using data-processing inequalities, and conjectures optimal strategies.

Findings

Derived lower bounds using Dictator and Majority functions.

Established upper bounds via entropy and Fourier analysis.

Extended bounds to general channels with data-processing inequalities.

Abstract

What is the value of a single bit to a guesser? We study this problem in a setup where Alice wishes to guess an i.i.d. random vector, and can procure one bit of information from Bob, who observes this vector through a memoryless channel. We are interested in the guessing efficiency, which we define as the best possible multiplicative reduction in Alice's guessing-moments obtainable by observing Bob's bit. For the case of a uniform binary vector observed through a binary symmetric channel, we provide two lower bounds on the guessing efficiency by analyzing the performance of the Dictator and Majority functions, and two upper bounds via maximum entropy and Fourier-analytic / hypercontractivity arguments. We then extend our maximum entropy argument to give a lower bound on the guessing efficiency for a general channel with a binary uniform input, via the strong data-processing inequality constant of the reverse channel. We compute this bound for the binary erasure channel, and conjecture that Greedy Dictator functions achieve the guessing efficiency.