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This paper studies MMSE estimation on the basis of quantized noisy observations. It presents nonasymptotic bounds on MMSE regret due to quantization for two settings: (1) estimation of a scalar random variable given a quantized vector of n conditionally independent observations, and (2) estimation of a p-dimensional random vector given a quantized vector of n observations (not necessarily independent) when the full MMSE estimator has a subgaussian concentration property.

On MMSE estimation from quantized observations in the nonasymptotic regime