Estimation of non-Gaussian random variables in Gaussian noise: Properties of the MMSE

This work studies the properties of the minimum mean-square error (MMSE) of estimating an arbitrary random variable contaminated by Gaussian noise based on the observation. The MMSE can be regarded as a function of the signal-to-noise ratio (SNR), as well as a functional or transform of the input distribution. This paper shows that the MMSE is analytic in SNR for every random variable. Simple expressions for the derivatives of the MMSE as a function of the SNR are obtained. Since the input-output mutual information can be written as the integral of the MMSE as a function of SNR, the results also lead to higher derivatives of the mutual information. The MMSE and mutual informationpsilas convexity in the SNR and concavity in the input distribution are established. It is shown that there can be only one SNR for which the MMSE of a Gaussian random variable and that of a non-Gaussian random variable coincide. Application of the properties of the MMSE to the scalar Gaussian broadcast channel problem is presented.