基于非线性期望理论的分布不确定性下香农信源编码定理再探讨
Revisiting Shannon's Source Coding Theorem with Distributional Uncertainty under the Nonlinear Expectation Theory
AI总结：
本文基于非线性期望理论，针对分布不确定的信源建立非线性信源编码定理，推广了信息熵概念，为通信网络中信源的分析提供了更全面的理论框架。
AI中文摘要：
在经典信息论中，信源由单一、精确已知的概率分布建模。然而，在充满意外、非平稳和异质随机事件的日益复杂的通信网络中，用精确且定义明确的概率分布描述随机变量的假设显得有些理想化。因此，在信息论中，为分析信源而放松确定性概率模型的假设，刻画信源消息分布的不确定性十分重要。本文基于扩展经典概率论的非线性期望理论（一种新的公理体系），研究分布本身不确定的信源，将其称为不确定分布信源。我们将基本概念信息熵推广为非线性信息熵，用于衡量不确定分布信源包含的信息量。利用次线性期望下的强大数定律，我们建立了非线性信源编码定理，该定理不仅表明在最大错误概率准则下，非线性信息熵是不确定分布信源可实现编码率下确界的上界，还确定了在最小错误概率准则下，不确定分布信源编码率的一个聚点。我们的发现表明，引入非线性期望理论可让人们对信源有更全面的理解。
英文摘要：
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.
