Differential Entropy of the Conditional Expectation under Additive Gaussian Noise

TL;DR

This paper derives new bounds on the differential entropy of the conditional mean in Gaussian noise channels, with implications for estimation theory and information bounds.

Contribution

It introduces a novel lower bound on the differential entropy of the conditional mean for Gaussian noise models, extending to vector channels and exponential families.

Findings

New lower bound on differential entropy of the conditional mean

Extensions to vector Gaussian channels and exponential families

Applications to remote-source coding and CEO problems

Abstract

The conditional mean is a fundamental and important quantity whose applications include the theories of estimation and rate-distortion. It is also notoriously difficult to work with. This paper establishes novel bounds on the differential entropy of the conditional mean in the case of finite-variance input signals and additive Gaussian noise. The main result is a new lower bound in terms of the differential entropies of the input signal and the noisy observation. The main results are also extended to the vector Gaussian channel and to the natural exponential family. Various other properties such as upper bounds, asymptotics, Taylor series expansion, and connection to Fisher Information are obtained. Two applications of the lower bound in the remote-source coding and CEO problem are discussed.