Title: Simultaneous Inference for Multiple Proportions: A Multivariate Beta-Binomial Model Show Chinese title

Title: 多个比例的同时推断：多元Beta-二项式模型

Abstract: Statistical inference in high-dimensional settings is challenging when standard unregularized methods are employed. In this work, we focus on the case of multiple correlated proportions for which we develop a Bayesian inference framework. For this purpose, we construct an $m$-dimensional Beta distribution from a $2^m$-dimensional Dirichlet distribution, building on work by Olkin and Trikalinos (2015). This readily leads to a multivariate Beta-binomial model for which simple update rules from the common Dirichlet-multinomial model can be adopted. From the frequentist perspective, this approach amounts to adding pseudo-observations to the data and allows a joint shrinkage estimation of mean vector and covariance matrix. For higher dimensions ($m > 10$), the extensive model based on $2^m$ parameters starts to become numerically infeasible. To counter this problem, we utilize a reduced parametrisation which has only $1 + m(m + 1)/2$ parameters describing first and second order moments. A copula model can then be used to approximate the (posterior) multivariate Beta distribution. A natural inference goal is the construction of multivariate credible regions. The properties of different credible regions are assessed in a simulation study in the context of investigating the accuracy of multiple binary classifiers. It is shown that the extensive and copula approach lead to a (Bayes) coverage probability very close to the target level. In this regard, they outperform credible regions based on a normal approximation of the posterior distribution, in particular for small sample sizes. Additionally, they always lead to credible regions which lie entirely in the parameter space which is not the case when the normal approximation is used. Show Chinese abstract

Abstract: 在使用标准未正则化方法时，高维设置下的统计推断具有挑战性。 在这项工作中，我们专注于多个相关比例的情况，并为此开发了一个贝叶斯推断框架。 为此，我们基于 Olkin 和 Trikalinos (2015) 的工作，从一个$2^m$维的狄利克雷分布构建了一个$m$维的 Beta 分布。 这自然引出了一个多元 Beta-二项式模型，可以采用常见的狄利克雷-多项式模型的简单更新规则。 从频率学派的角度来看，这种方法相当于向数据添加伪观测值，并允许联合收缩估计均值向量和协方差矩阵。 对于更高的维度（$m > 10$），基于$2^m$参数的广泛模型开始变得数值上不可行。 为了解决这个问题，我们利用了一种减少参数化的形式，该形式仅描述了前两阶矩，共有$1 + m(m + 1)/2$个参数。 然后可以使用 Copula 模型来近似后验多元 Beta 分布。 一个自然的推断目标是构造多变量可信区域。 在调查多个二元分类器准确性的情境下，通过模拟研究评估了不同可信区域的属性。 结果显示，广泛的和 Copula 方法得到的（贝叶斯）覆盖概率非常接近目标水平。 在这方面，它们优于基于后验分布正态近似的可信区域，特别是在样本量较小时。 此外，它们始终生成完全位于参数空间内的可信区域，而当使用正态近似时则不然。