Tight MMSE Bounds for the AGN Channel Under KL Divergence Constraints on\n the Input Distribution

Tight bounds on the minimum mean square error for the additive Gaussian noise\nchannel are derived, when the input distribution is constrained to be\nepsilon-close to a Gaussian reference distribution in terms of the\nKullback--Leibler divergence. The distributions that attain the bounds are\nshown be Gaussian whose means are identical to that of the reference\ndistribution and whose covariance matrices are defined implicitly via systems\nof matrix equations. The estimator that attains the upper bound is identified\nas a minimax optimal estimator that is robust against deviations from the\nassumed prior. The lower bound is shown to provide a potentially tighter\nalternative to the Cramer--Rao bound. Both properties are illustrated with\nnumerical examples.\n