Communications in Information and Systems

Volume 9 (2009)

Number 2

A Novel Information Transmission Problem and its Optimal Solution

Pages: 141 – 162

DOI: http://dx.doi.org/10.4310/CIS.2009.v9.n2.a1

Authors

Eric Bach

Jin-Yi Cai

Abstract

We propose and study a new information transmission problem motivated by today's internet. Suppose a real number needs to be transmitted in a network. This real number may represent data or control and pricing information of the network. We propose a new transmission model in which the real number is encoded using Bernoulli trials. This differs from the traditional framework of Shannon's information theory. We propose a natural criterion for the quality of an encoding scheme. Choosing the best encoding reduces to a problem in the calculus of variations, which we solve rigorously. In particular, we show there is a unique optimal encoding, and give an explicit formula for it.

We also solve the problem in a more general setting in which there is prior information about the real number, or a desire to weight errors for different values non-uniformly.

Our tools come mainly from real analysis and measure-theoretic probability. We also explore a connection to classical mechanics.

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