Resolution Limits of Noisy 20 Questions Estimation
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We establish fundamental limits on estimation accuracy for the noisy 20 questions problem with measurement dependent noise and introduce optimal non-adaptive procedures that achieve this limits. The minimal achievable resolution is defined as the absolute difference between the estimated and the true values of the target random variable, given a finite number of queries constrained by the excess-resolution probability. Inspired by the relationship between the 20 questions problem and the channel coding problem, we derive non-asymptotic bounds on the minimal achievable resolution. Furthermore, applying the Berry-Esseen theorem to our non-asymptotic bounds, we obtain a second-order asymptotic approximation to finite blocklength performance, specifically the achievable resolution of optimal non-adaptive query procedures with a finite number of queries subject to the excess-resolution probability constraint. We specialize our second-order results to measurement dependent versions of several channel models including the binary symmetric, the binary erasure and the binary Z- channels. Our results are then extended to adaptive query procedures, establishing a lower bound on the resolution gain associated with adaptive querying.
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