Revisiting Shannon's Source Coding Theorem with Distributional Uncertainty under the Nonlinear Expectation Theory

2026-08-17

AI Summary

This study addresses the failure of classical source coding theorems caused by distributional uncertainty in complex networks. Leveraging nonlinear expectation theory and the law of large numbers under sublinear expectations, this work establishes an axiomatic framework and introduces the concept of nonlinear information entropy. On this basis, a source coding theorem under distributional uncertainty is formulated to characterize achievable rate bounds. Specifically, it is proven that nonlinear information entropy serves as the upper bound on coding rates under the maximum error criterion and as a cluster point under the minimum error criterion. These findings overcome the limitations of traditional frameworks and significantly advance the theoretical understanding of encoding mechanisms for sources with uncertain distributions.

Abstract

In classical information theory, a source is modeled by a single, precisely known probability distribution. However, in the increasingly complex communication networks full of unanticipated, nonstationary, and heterogeneous random events, the assumption of precise and well-defined probability distributions to describe random variables appears somewhat idealized. Therefore, it is important to characterize the uncertainty of distributions of source messages, subject to relaxing the assumption of deterministic probability models for analyzing information sources in information theory. Based on the nonlinear expectation theory, a novel axiomatical system that extends classical probability theory, this paper investigates the information sources whose distributions themselves are uncertain, and refers to them as uncertain-distribution sources. We generalize the fundamental concept information entropy to nonlinear information entropy, which describes the measurement of the amount of information contained in a uncertain-distribution source. By using the strong law of large numbers under sublinear expectation, we establish a nonlinear source coding theorem, which not only shows that the nonlinear information entropy is the upper bound for the infimum of achievable coding rate of uncertain-distribution sources under the maximum error probability criterion, but also determines a cluster point of the coding rate of uncertain-distribution sources under the minimum error probability criterion. Our findings reveal that the introduction of nonlinear expectation theory allows for a more comprehensive understanding of information sources.

Problem

• Distributional Uncertainty
• Nonlinear Expectation Theory
• Source Coding Theorem
• Uncertain-distribution Sources
• Nonlinear Information Entropy

Innovation

• Nonlinear Expectation Theory
• Uncertain-Distribution Sources
• Nonlinear Information Entropy
• Nonlinear Source Coding Theorem
• Distributional Uncertainty