  I present several new relations between mutual information (MI) and
statistical estimation error for a system that can be regarded simultaneously
as a communication channel and as an estimator of an input parameter. I first
derive a second-order result between MI and Fisher information (FI) that is
valid for sufficiently narrow priors, but arbitrary channels. A second relation
furnishes a lower bound on the MI in terms of the minimum mean-squared error
(MMSE) on the Bayesian estimation of the input parameter from the channel
output, one that is valid for arbitrary channels and priors. The existence of
such a lower bound, while extending previous work relating the MI to the FI
that is valid only in the asymptotic and high-SNR limits, elucidates further
the fundamental connection between information and estimation theoretic
measures of fidelity. The remaining relations I present are inequalities and
correspondences among MI, FI, and MMSE in the presence of nuisance parameters.
