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.