# Mean Estimation from One-Bit Measurements

We consider the problem of estimating the mean of a symmetric log-concave distribution under the following constraint: only a single bit per sample from this distribution is available to the estimator. We study the mean squared error (MSE) risk in this estimation as a function of the number of samples, and hence the number of bits, from this distribution. Under an adaptive setting in which each bit is a function of the current sample and the previously observed bits, we show that the optimal relative efficiency compared to the sample mean is the efficiency of the median. For example, in estimating the mean of a normal distribution, a constraint of one bit per sample incurs a penalty of π/2 in sample size compared to the unconstrained case. We also consider a distributed setting where each one-bit message is only a function of a single sample. We derive lower bounds on the MSE in this setting, and show that the optimal efficiency can only be attained at a finite number of points in the parameter space. Finally, we analyze a distributed setting where the bits are obtained by comparing each sample against a prescribed threshold. Consequently, we consider the threshold density that minimizes the maximal MSE. Our results indicate that estimating the mean from one-bit measurements is equivalent to estimating the sample median from these measurements. In the adaptive case, this estimate can be done with vanishing error for any point in the parameter space. In the distributed case, this estimate can be done with vanishing error only for a finite number of possible values for the unknown mean.

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