| Summary: | 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 <i>k</i> information bits from Bob, who has observed this vector through a memoryless channel. We are interested in the <i>guessing ratio</i>, 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>) and majority functions (for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>). We further provide a lower bound via maximum entropy (for general <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>) and a lower bound based on Fourier-analytic/hypercontractivity arguments (for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>). 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.
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