A Bivariate Beta Distribution with Arbitrary Beta Marginals and its Generalization to a Correlated Dirichlet Distribution

We discuss a bivariate beta distribution that can model arbitrary beta-distributed marginals with a positive correlation. The distribution is constructed from six independent gamma-distributed random variates. We show how the parameters of the distribution can be fit to data using moment matching. Previous work used an approximate and sometimes inaccurate method to compute the covariance. Here, we derive all product moments and the exact covariance, which can easily be computed numerically. The bivariate case can be generalized to a multivariate distribution with arbitrary beta-distributed marginals. Furthermore, we generalize the distribution from two marginal beta to two marginal Dirichlet distributions. The resulting correlated Dirichlet distribution makes it possible to model two correlated Dirichlet-distributed random vectors.