  Tight bounds on the minimum mean square error for the additive Gaussian noise
channel are derived, when the input distribution is constrained to be
epsilon-close to a Gaussian reference distribution in terms of the
Kullback--Leibler divergence. The distributions that attain the bounds are
shown be Gaussian whose means are identical to that of the reference
distribution and whose covariance matrices are defined implicitly via systems
of matrix equations. The estimator that attains the upper bound is identified
as a minimax optimal estimator that is robust against deviations from the
assumed prior. The lower bound is shown to provide a potentially tighter
alternative to the Cramer--Rao bound. Both properties are illustrated with
numerical examples.
