Uncoded transmission is exactly optimal for a simple Gaussian "sensor" network

One of the simplest sensor network models has one single underlying Gaussian source of interest, observed by many sensors, subject to independent Gaussian observation noise. The sensors communicate over a standard Gaussian multiple-access channel to a fusion center whose goal is to estimate the underlying source with respect to mean-squared error. In this note, a theorem of Witsenhausen is shown to imply that an optimal communication strategy is uncoded transmission, i.e., each sensors' channel input is merely a scaled version of its noisy observation.