Simultaneous Inference for Multiple Proportions: A Multivariate Beta-Binomial Model
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In this work, the construction of an m-dimensional Beta distribution from a 2^m-dimensional Dirichlet distribution is proposed, extending work by Olkin and Trikalinos (2015). To illustrate for which correlation structures such a distribution can exist, a necessary and sufficient condition is derived. This readily leads to a multivariate Beta-Binomial model for which simple update rules from the common Dirichlet-multinomial model can be adopted. A natural inference goal is the construction of multivariate credible regions. This is for instance possible by sampling from the underlying Dirichlet distribution. For higher dimensions (m>10), this extensive approach starts to become numerically infeasible. To counter this problem, a reduced representation is proposed which has only 1+m(m+1)/2 parameters describing first and second order moments. A copula approach can then be used to obtain a credible region. 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.
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