Estimation in Gaussian Noise: Properties of the Minimum Mean-Square Error

TL;DR

This paper investigates the mathematical properties of the MMSE in Gaussian noise, revealing its concavity, differentiability, and the behavior of its derivatives, which have implications for information theory and channel capacity proofs.

Contribution

It establishes the concavity, differentiability, and analytic properties of MMSE as a function of SNR and input distribution, providing new proofs for key information theory results.

Findings

MMSE is concave in input distribution at fixed SNR

MMSE is infinitely differentiable and real analytic in SNR for given input

Curves of MMSE for Gaussian and non-Gaussian inputs cross at most once

Abstract

Consider the minimum mean-square error (MMSE) of estimating an arbitrary random variable from its observation contaminated by Gaussian noise. The MMSE can be regarded as a function of the signal-to-noise ratio (SNR) as well as a functional of the input distribution (of the random variable to be estimated). It is shown that the MMSE is concave in the input distribution at any given SNR. For a given input distribution, the MMSE is found to be infinitely differentiable at all positive SNR, and in fact a real analytic function in SNR under mild conditions. The key to these regularity results is that the posterior distribution conditioned on the observation through Gaussian channels always decays at least as quickly as some Gaussian density. Furthermore, simple expressions for the first three derivatives of the MMSE with respect to the SNR are obtained. It is also shown that, as functions of the SNR, the curves for the MMSE of a Gaussian input and that of a non-Gaussian input cross at most once over all SNRs. These properties lead to simple proofs of the facts that Gaussian inputs achieve both the secrecy capacity of scalar Gaussian wiretap channels and the capacity of scalar Gaussian broadcast channels, as well as a simple proof of the entropy power inequality in the special case where one of the variables is Gaussian.