  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.
