In this paper, we consider a Bayesian estimation problem in a sensor network where the local sensor observations are quantized before their transmission to the fusion center (FC). Inspired by Widrow's statistical theory on quantization, at the FC, instead of fusing the quantized data directly, we propose to fuse the post-processed data obtained by adding independent controlled noise to the received quantized data. The injected noise acts like a low-pass filter in the characteristic function (CF) domain such that the output is an approximation of the original raw observation. The optimal minimum mean squared error (MMSE) estimator and the posterior Cramér-Rao lower bound for this estimation problem are derived. Based on the Fisher information, the optimal controlled Gaussian noise and the optimal bit allocation are obtained. In addition, a near-optimal linear MMSE estimator is derived to reduce the computational complexity significantly.