从信息数据中学习你的熵：一个确保一致识别广义熵的公理
Learn your entropy from informative data: an axiom ensuring the consistent identification of generalized entropies
AI总结：
本文提出一个公理，确保从信息数据中一致识别广义熵，通过排除无信息分布下的熵参数，验证了Rényi熵的唯一性，并扩展了最大似然原理在估计熵参数中的应用。
摘要：
Shannon熵作为信息论、统计物理和推断方法的核心，通过Shannon-Khinchin或Shore-Johnson公理唯一确定。广义Shannon熵的推广，受非扩展性或非平衡系统研究启发，放松部分公理，导致以特定`熵`参数索引的熵族。通常，这些参数的选择需要系统先验知识或导致不一致。本文引入一个简单公理：任何熵族中，无信息（均匀）概率分布无法推断熵参数。当应用于Uffink-Jizba-Korbel和Hanel-Thurner熵时，该公理仅选择Rényi熵为可行。它还扩展了最大似然原理的一致性，可从数据中估计熵参数。在广义最大熵框架中，该公理表明最大化对数似然总是等于负Shannon熵，即使推断的概率分布最大化广义熵而非Shannon熵，解决了先前方法中的多个问题。
英文摘要：
Shannon entropy, a cornerstone of information theory, statistical physics and inference methods, is uniquely identified by the Shannon-Khinchin or Shore-Johnson axioms. Generalizations of Shannon entropy, motivated by the study of non-extensive or non-ergodic systems, relax some of these axioms and lead to entropy families indexed by certain `entropic' parameters. In general, the selection of these parameters requires pre-knowledge of the system or encounters inconsistencies. Here we introduce a simple axiom for any entropy family: namely, that no entropic parameter can be inferred from a completely uninformative (uniform) probability distribution. When applied to the Uffink-Jizba-Korbel and Hanel-Thurner entropies, the axiom selects only Rényi entropy as viable. It also extends consistency with the Maximum Likelihood principle, which can then be generalized to estimate the entropic parameter purely from data, as we confirm numerically. Remarkably, in a generalized maximum-entropy framework the axiom implies that the maximized log-likelihood always equals minus Shannon entropy, even if the inferred probability distribution maximizes a generalized entropy and not Shannon's, solving a series of problems encountered in previous approaches.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
