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This paper proposes a new family of lower and upper bounds on the minimummean squared error (MMSE). The key idea is to minimize/maximize the MMSEsubject to the constraint that the joint distribution of the input-outputstatistics lies in a Kullback-Leibler divergence ball centered at some Gaussianreference distribution. Both bounds are tight and are attained by Gaussiandistributions whose mean is identical to that of the reference distribution andwhose covariance matrix is determined by a scalar parameter that can beobtained by finding the root of a monotonic function. The upper boundcorresponds to a minimax optimal estimator and provides performance guaranteesunder distributional uncertainty. The lower bound provides an alternative towell-known inequalities in estimation theory, such as the Cram\'er-Rao bound,that is potentially tighter and defined for a larger class of distributions.Examples of applications in signal processing and information theory illustratethe usefulness of the proposed bounds in practice.

MMSE Bounds Under Kullback-Leibler Divergence Constraints on the Joint Input-Output Distribution