熵泛函的结构刻画
A Structural Characterization of Entropy Functionals
AI总结：
该研究通过测度论框架刻画熵泛函的容许性，解决雷尼提出的广义均值替代问题，区分出雷尼族与香农熵，还得到新的容许熵和散度族。
AI中文摘要：
熵泛函及其相关散度是最大熵推理、最小散度估计、拟合优度检验等诸多统计方法的基础，但在香农（Shannon）、雷尼（Rényi）、 Tsallis 熵及更一般的熵之间进行选择，往往是出于惯例而非结构原则。我们引入一种测度论框架，其中容许性要求：当输入测度相对于参考测度绝对连续时，输入测度的熵需上界于参考测度的熵。在广义均值值复合下，我们刻画了所有此类熵泛函，得到一个由均值生成元、熵尺度和可加性假设依次决定的四级层级。连续严格单调生成元 $g$ 为容许的充要条件是：当 $g$ 递增时，映射 $t\mapsto g(1/t)$ 严格凸；当 $g$ 递减时，映射 $t\mapsto g(1/t)$ 严格凹。这解决了雷尼（Proc. 4th Berkeley Sympos. Math. Statist. Prob., 1961）提出的问题：在其熵公理化中，哪些广义均值可替代算术均值。该准则等价于相关 Csiszár $f$-散度生成元的严格凸性，因此在马尔可夫核下产生具有严格等式条件的数据处理。在该层级中，乘积可加性区分出雷尼族，而内部可加性（或乘积可加性结合算术均值值复合）区分出香农熵。该刻画具有构造性，且产生新的容许熵与散度族，包括积分变换示例。
英文摘要：
Entropy functionals and their associated divergences underlie many statistical methods, including maximum entropy inference, minimum divergence estimation, and goodness-of-fit testing, yet choosing among Shannon, Rényi, Tsallis, and more general entropies is often a matter of convention rather than structural principle. We introduce a measure theoretic framework in which admissibility requires the entropy of an input measure to be bounded above by that of its reference measure whenever the former is absolutely continuous with respect to the latter. Under generalized mean-value composition, we characterize all such entropy functionals and obtain a four-level hierarchy determined successively by the mean generator, entropy scale, and additivity assumptions. A continuous strictly monotone generator $g$ is admissible exactly when $t\mapsto g(1/t)$ is strictly convex for increasing $g$, or strictly concave for decreasing $g$. This resolves a question posed by Rényi (Proc. 4th Berkeley Sympos. Math. Statist. Prob., 1961) concerning which generalized means may replace the arithmetic mean in his entropy axiomatization. The same criterion is equivalent to strict convexity of an associated Csiszár $f$-divergence generator and therefore yields data processing under Markov kernels with an exact equality condition. Within this hierarchy, product additivity singles out the Rényi family, while internal additivity, or product additivity together with arithmetic mean-value composition, singles out Shannon entropy. The characterization is constructive and yields new admissible entropy and divergence families, including integral-transform examples.
