The coupled entropy, $H_κ,$ is proven to uniquely satisfy the requirement that a generalized entropy be a measure of the uncertainty at the scale, $σ,$ for a class of non-exponential distributions. The coupled stretched exponential distributions, including the generalized Pareto and Student's t distributions, are uniquely parameterized to quantify linear uncertainty with the scale and nonlinear uncertainty with the tail shape for a broad class of complex systems. Thereby, the coupled entropy optimizes the representation of the uncertainty due to linear sources. Lemmas for the composability and extensivity of the coupled entropy are proven. The uniqueness of the coupled entropy is further supported by demonstrating consistent thermodynamic relationships, which correspond to a model used for the momentum of high-energy particle collisions. Applications of the coupled entropy in measuring statistical complexity, training variational inference algorithms, and designing communication channels are reviewed.

翻译：耦合熵$H_κ$被证明是唯一满足以下要求的广义熵：对于一类非指数分布，该熵是尺度$σ$下不确定性的度量。耦合拉伸指数分布（包括广义帕累托分布和学生t分布）被唯一参数化，用以量化广泛复杂系统中由尺度决定的线性不确定性和由尾部形态决定的非线性不确定性。因此，耦合熵优化了对线性源所产生不确定性的表征。本文证明了耦合熵的可组合性与广延性引理。通过展示其与高能粒子碰撞动量模型相对应的、一致的热力学关系，进一步支持了耦合熵的唯一性。文中综述了耦合熵在测量统计复杂性、训练变分推断算法以及设计通信信道等方面的应用。