  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.
