A Nonstochastic Information Theory for Communication and State Estimation

TL;DR

This paper develops a nonstochastic information theory framework to analyze communication and state estimation without relying on probability, introducing a maximin information functional that aligns with classical channel capacity and enables robust control applications.

Contribution

It introduces a novel nonstochastic information measure and demonstrates its equivalence to zero-error capacity, bridging control and communication theories without probabilistic assumptions.

Findings

Maximin information rate equals zero-error channel capacity.

Provides tight conditions for state estimation over error-prone channels.

Bridges stochastic and nonstochastic frameworks in control and communication.

Abstract

In communications, unknown variables are usually modelled as random variables, and concepts such as independence, entropy and information are defined in terms of the underlying probability distributions. In contrast, control theory often treats uncertainties and disturbances as bounded unknowns having no statistical structure. The area of networked control combines both fields, raising the question of whether it is possible to construct meaningful analogues of stochastic concepts such as independence, Markovness, entropy and information without assuming a probability space. This paper introduces a framework for doing so, leading to the construction of a maximin information functional for nonstochastic variables. It is shown that the largest maximin information rate through a memoryless, error-prone channel in this framework coincides with the block-coding zero-error capacity of the channel. Maximin information is then used to derive tight conditions for uniformly estimating the state of a linear time-invariant system over such a channel, paralleling recent results of Matveev and Savkin.