基尼-辛普森指数的非参数贝叶斯推断
Nonparametric Bayesian inference for the Gini-Simpson index
AI总结：
研究物种分布等离散标记量分析中，针对基尼-辛普森指数，比较几种非参数先验模型，指出传统对称狄利克雷分布的问题，提出改进规范，对比经典弗格森狄利克雷过程与泊松-狄利克雷模型，给出后验均值表示并经渐近分析等验证。
AI中文摘要：
许多统计问题涉及物种分布或更一般的离散标记量的分析。评估物种多样性是理解种群结构的关键步骤，基尼-辛普森指数是应用最广泛的多样性度量之一。本文研究了几种成熟的物种频率非参数先验模型，并将它们与该框架内新提出的分布进行比较。具体而言，我们证明传统对称狄利克雷分布在期望和离散度方面存在某些不良性质。采用参数依赖于物种数量的替代对称狄利克雷规范可缓解这些限制。这种修改后的公式具有分析易处理性和主要摘要的可解释性。此外，当不同物种数量无限时，经典弗格森狄利克雷过程与更一般的泊松-狄利克雷模型相比表现不佳。值得注意的是，在后一种模型中，多样性指数的后验均值可表示为最优经典无偏估计量和先验期望的凸组合。理论结果通过渐近分析得到进一步支持，并与经典对应结果进行系统比较。
英文摘要：
Many statistical problems concern the analysis of species distributions or, more generally, of discrete labeled quantities. Assessing species diversity constitutes a key step toward understanding population structure, and the Gini-Simpson index is among the most widely adopted diversity measures. In this manuscript, we examine several well-established nonparametric prior models for species frequencies and compare them with a newly proposed distribution within this framework. Specifically, we demonstrate that the conventional symmetric Dirichlet distribution leads to certain undesirable properties in terms of expectation and dispersion. These limitations can be mitigated by adopting an alternative symmetric Dirichlet specification, in which the parameter depends on the number of species. This modified formulation is characterized by analytical tractability and interpretability of its main summaries. Furthermore, when the number of distinct species is infinite, the classical Ferguson Dirichlet process exhibits unsatisfactory behavior compared to the more general Poisson-Dirichlet model. Notably, within this latter model, the posterior mean of the diversity index can be expressed as a convex combination of the optimal classical unbiased estimator and the prior expectation. Theoretical results are further supported by asymptotic analyses and systematically compared with their classical counterparts.
