Guessing with a bit of help
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entropy-22-00039.pdf
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544.4 KB
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7b7528d8044c666065a7f5d6b94d6248
Author(s)
Weinberger, Nir
Shayevitz, Ofer
Date Issued
December 2019
Journal
Entropy
Publisher
MDPI
Citation
Weinberger, Nir, and Ofer Shayevitz, "Guessing with a bit of help." Entropy 22, 1 (Dec. 2019): no. 39 doi 10.3390/e22010039 ©2019 Author(s)
Version
Final published version
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. ©2019
MIT Department
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
Massachusetts Institute of Technology. Institute for Data, Systems, and Society
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Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.3390/e22010039
