Simultaneous Inference for Multiple Proportions: A Multivariate Beta-Binomial Model

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.