  Consider the minimum mean-square error (MMSE) of estimating an arbitrary
random variable from its observation contaminated by Gaussian noise. The MMSE
can be regarded as a function of the signal-to-noise ratio (SNR) as well as a
functional of the input distribution (of the random variable to be estimated).
It is shown that the MMSE is concave in the input distribution at any given
SNR. For a given input distribution, the MMSE is found to be infinitely
differentiable at all positive SNR, and in fact a real analytic function in SNR
under mild conditions. The key to these regularity results is that the
posterior distribution conditioned on the observation through Gaussian channels
always decays at least as quickly as some Gaussian density. Furthermore, simple
expressions for the first three derivatives of the MMSE with respect to the SNR
are obtained. It is also shown that, as functions of the SNR, the curves for
the MMSE of a Gaussian input and that of a non-Gaussian input cross at most
once over all SNRs. These properties lead to simple proofs of the facts that
Gaussian inputs achieve both the secrecy capacity of scalar Gaussian wiretap
channels and the capacity of scalar Gaussian broadcast channels, as well as a
simple proof of the entropy power inequality in the special case where one of
the variables is Gaussian.
