arXiv2026
A mathematical framework for information-theoretic analysis is established, with a new viewpoint of describing transmitted messages and communication channels by the nonlinear expectation theory, beyond classical probability framework. Based on nonlinear expectation theory, in this paper we define several fundamental concepts, such as nonlinear information entropy, nonlinear joint entropy, and nonlinear mutual information, and establish their basic properties. By using strong law of large numbers under sublinear expectations, we propose a nonlinear source coding theorem, which shows that nonlinear entropy is a sufficient achievability threshold for the achievable coding rate of uncertain-distribution sources under the maximum error probability criterion, and determines a cluster point of the achievable coding rate under the minimum error probability criterion. We also propose a nonlinear channel coding theorem, providing explicit expressions of the upper bound and a cluster point under the maximum and minimum error probability criterion, respectively, for the achievable coding rate of uncertain-distribution channels. Additionally, we propose a nonlinear rate-distortion source coding theorem, proving that the rate-distortion function based on nonlinear mutual information is a cluster point of the lossy compression performance of uncertain-distribution sources under the minimum expected distortion criterion. Finally, we show some examples and simulations to consolidate our theoretic study.