MMSE Bounds for Additive Noise Channels Under Kullback–Leibler Divergence Constraints on the Input Distribution

Upper and lower bounds on the minimum mean square error for additive noise channels are derived when the input distribution is constrained to be close to a Gaussian reference distribution in terms of the Kullback-Leibler divergence. The upper bound is tight and is attained by a Gaussian distribution whose mean is identical to that of the reference distribution and whose covariance matrix is defined implicitly via a system of non-linear 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 provides an alternative to well-known inequalities in estimation and information theory-such as the Cramér-Rao lower bound, Stam's inequality, or the entropy power inequality-that is potentially tighter and defined for a larger class of input distributions. Several examples of applications in signal processing and information theory illustrate the usefulness of the proposed bounds in practice.