## 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.

Original language | English |
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Pages (from-to) | 39 |

Number of pages | 1 |

Journal | Entropy |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2020 |

## Keywords

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