  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.
