Relations between Information and Estimation in Discrete-Time L\'evy Channels

TL;DR

This paper extends fundamental information-estimation relations from Gaussian and Poisson channels to a broader class called Le9vy channels, which include gamma and negative binomial channels, unifying and generalizing previous results.

Contribution

It introduces Le9vy channels with infinitely divisible output distributions and establishes new relations between mutual information and estimation loss for these channels.

Findings

Relations hold for gamma and negative binomial channels

Unified framework for information-estimation in Le9vy channels

Extensions to mismatched estimation scenarios

Abstract

Fundamental relations between information and estimation have been established in the literature for the discrete-time Gaussian and Poisson channels. In this work, we demonstrate that such relations hold for a much larger class of observation models. We introduce the natural family of discrete-time L\'evy channels where the distribution of the output conditioned on the input is infinitely divisible. For L\'evy channels, we establish new representations relating the mutual information between the channel input and output to an optimal expected estimation loss, thereby unifying and considerably extending results from the Gaussian and Poisson settings. We demonstrate the richness of our results by working out two examples of L\'evy channels, namely the gamma channel and the negative binomial channel, with corresponding relations between information and estimation. Extensions to the setting of mismatched estimation are also presented.