In recent years, biodiversity measures have gained prominence as essential tools for ecological and environmental assessments, particularly in the context of increasingly complex and large-scale datasets. We provide a comprehensive review of diversity measures, including the Gini-Simpson index, Hill numbers, and Rao's quadratic entropy, examining their roles in capturing various aspects of biodiversity. Among these, Rao's quadratic entropy stands out for its ability to incorporate not only species abundance but also functional and genetic dissimilarities. The paper emphasizes the statistical and ecological significance of Rao's quadratic entropy under the information geometry framework. We explore the distribution maximizing such a diversity measure under linear constraints that reflect ecological realities, such as resource competition or habitat suitability. Furthermore, we discuss a unified approach of the Leinster-Cobbold index combining Hill numbers and Rao's entropy, allowing for an adaptable and similarity-sensitive measure of biodiversity. Finally, we discuss the information geometry associated with the maximum diversity distribution focusing on the cross diversity measures such as the cross-entropy.

翻译：近年来，生物多样性度量作为生态与环境评估的关键工具日益受到重视，尤其是在处理日益复杂和大规模数据集时。本文系统综述了包括基尼-辛普森指数、希尔数以及Rao二次熵在内的多样性度量方法，探讨它们在捕捉生物多样性不同维度中的作用。其中，Rao二次熵因其不仅能纳入物种丰度，还能整合功能与遗传差异的特性而尤为突出。本文重点阐释了在信息几何框架下Rao二次熵的统计学与生态学意义。我们探究了在反映生态现实（如资源竞争或栖息地适宜性）的线性约束条件下，使此类多样性度量最大化的分布规律。此外，我们讨论了整合希尔数与Rao熵的Leinster-Cobbold指数的统一框架，该框架可形成适应性强且对相似性敏感的多样性度量方法。最后，我们聚焦于交叉熵等交叉多样性度量，探讨了与最大多样性分布相关的信息几何结构。