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Certain Relations between Mutual Information and Fidelity of Statistical Estimation

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.