@article{81449f07dd424549aedadde6aa92d7bd,

title = "Guessing with a bit of help",

abstract = "What is the value of just a few bits to a guesser? We study this problem in a setup where Alice wishes to guess an independent and identically distributed (i.i.d.) random vector and can procure a fixed number of k information bits from Bob, who has observed this vector through a memoryless channel. We are interested in the guessing ratio, which we define as the ratio of Alice's guessing-moments with and without observing Bob's bits. For the case of a uniform binary vector observed through a binary symmetric channel, we provide two upper bounds on the guessing ratio by analyzing the performance of the dictator (for general k ≥ 1) and majority functions (for k = 1). We further provide a lower bound via maximum entropy (for general k ≥ 1) and a lower bound based on Fourier-analytic/hypercontractivity arguments (for k = 1). We then extend our maximum entropy argument to give a lower bound on the guessing ratio for a general channel with a binary uniform input that is expressed using 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 optimal guessing ratio.",

keywords = "Boolean functions, Fourier analysis, Guessing moments, Guessing with a helper, Hypercontractivity, Maximum entropy, Strong data-processing inequalities",

author = "Nir Weinberger and Ofer Shayevitz",

note = "Publisher Copyright: {\textcopyright} 2019 by the authors.",

year = "2020",

month = jan,

day = "1",

doi = "10.3390/e22010039",

language = "אנגלית",

volume = "22",

pages = "39",

journal = "Entropy",

issn = "1099-4300",

publisher = "MDPI Multidisciplinary Digital Publishing Institute",

number = "1",

}