  The conditional mean is a fundamental and important quantity whose
applications include the theories of estimation and rate-distortion. It is also
notoriously difficult to work with. This paper establishes novel bounds on the
differential entropy of the conditional mean in the case of finite-variance
input signals and additive Gaussian noise. The main result is a new lower bound
in terms of the differential entropies of the input signal and the noisy
observation. The main results are also extended to the vector Gaussian channel
and to the natural exponential family. Various other properties such as upper
bounds, asymptotics, Taylor series expansion, and connection to Fisher
Information are obtained. Two applications of the lower bound in the
remote-source coding and CEO problem are discussed.
