Correlated proportions appear in many real-world applications and present a unique challenge in terms of finding an appropriate probabilistic model due to their constrained nature. The bivariate beta is a natural extension of the well-known beta distribution to the space of correlated quantities on $[0, 1]^2$. Its construction is not unique, however. Over the years, many bivariate beta distributions have been proposed, ranging from three to eight or more parameters, and for which the joint density and distribution moments vary in terms of mathematical tractability. In this paper, we investigate the construction proposed by Olkin & Trikalinos (2015), which strikes a balance between parameter-richness and tractability. We provide classical (frequentist) and Bayesian approaches to estimation in the form of method-of-moments and latent variable/data augmentation coupled with Hamiltonian Monte Carlo, respectively. The elicitation of bivariate beta as a prior distribution is also discussed. The development of diagnostics for checking model fit and adequacy is explored in depth with the aid of Monte Carlo experiments under both well-specified and misspecified data-generating settings. Keywords: Bayesian estimation; bivariate beta; correlated proportions; diagnostics; method of moments.

翻译：相关比例出现在许多实际应用中，因其约束性质而给寻找合适的概率模型带来了独特挑战。双变量贝塔分布是经典贝塔分布向$[0, 1]^2$上相关量空间的自然扩展，但其构造方式并非唯一。多年来，学界提出了多种双变量贝塔分布，参数数量从三个到八个或更多不等，其联合密度和分布矩在数学可处理性方面各有差异。本文研究了Olkin & Trikalinos (2015)提出的构造方案，该方案在参数丰富性与可处理性之间取得了平衡。我们分别提供了经典（频率学派）和贝叶斯估计方法，前者采用矩估计法，后者采用潜变量/数据增广结合哈密顿蒙特卡洛法。同时讨论了将双变量贝塔作为先验分布的启发方法。通过蒙特卡洛实验，在数据生成过程正确设定与错误设定两种情形下，深入探讨了用于检验模型拟合优度与充分性的诊断方法开发。关键词：贝叶斯估计；双变量贝塔；相关比例；诊断；矩估计法