Differential entropy of the conditional expectation under additive gaussian voise

Date
2022
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Source Title
IEEE Transactions on Signal Processing
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Publisher
Institute of Electrical and Electronics Engineers
Volume
70
Issue
Pages
4851 - 4866
Language
English
Type
Article
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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.

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Keywords
Differential entropy, Conditional mean estimator, Gaussian noise, Exponential family, Remote source coding problem, CEO problem
Citation
Published Version (Please cite this version)