信息论在层次划分中的推广
Towards a generalization of information theory for hierarchical partitions
摘要：
复杂系统通常表现出覆盖广泛物理尺度的多层组织，因此研究其结构和功能的分层分解通常很方便。为了更好地理解这一现象，我们介绍了一个广义的信息论，与分层分区。我们开始重新审视最近推出的分层互信息（HMI），并表明它可以被写为一个经典的条件互信息条款的一级一级的总和。然后，我们证明了HMI是有界的从上到下由相应的层次联合熵。这样，在类比的经典情况下，我们得到许多其他经典的信息理论量的层次概括。特别是，我们证明，相对于其经典的对应，层次化的信息的变化的推广是不是一个度量距离，但它承认一个转换成一个。此外，着眼于潜在的应用现有的理论发展，我们展示了如何调整的HMI的机会。我们还证实和分析所有提出的理论结果与详尽的数值计算，并包括一个说明性的应用例子所介绍的形式主义。最后，我们提到了一些开放的问题，最终应解决拟议的推广信息论达到成熟。
英文摘要：
Complex systems often exhibit multiple levels of organization covering a wide range of physical scales, so the study of the hierarchical decomposition of their structure and function is frequently convenient. To better understand this phenomenon, we introduce a generalization of information theory that works with hierarchical partitions. We begin revisiting the recently introduced Hierarchical Mutual Information (HMI), and show that it can be written as a level by level summation of classical conditional mutual information terms. Then, we prove that the HMI is bounded from above by the corresponding hierarchical joint entropy. In this way, in analogy to the classical case, we derive hierarchical generalizations of many other classical information-theoretic quantities. In particular, we prove that, as opposed to its classical counterpart, the hierarchical generalization of the Variation of Information is not a metric distance, but it admits a transformation into one. Moreover, focusing on potential applications of the existing developments of the theory, we show how to adjust by chance the HMI. We also corroborate and analyze all the presented theoretical results with exhaustive numerical computations, and include an illustrative application example of the introduced formalism. Finally, we mention some open problems that should be eventually addressed for the proposed generalization of information theory to reach maturity.
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