  We derive the conjugate prior of the Dirichlet and beta distributions and
explore it with numerical examples to gain an intuitive understanding of the
distribution itself, its hyperparameters, and conditions concerning its
convergence. Due to the prior's intractability, we proceed to define and
analyze a closed-form approximation. Finally, we provide an algorithm
implementing this approximation that enables fully tractable Bayesian conjugate
treatment of Dirichlet and beta likelihoods without the need for Monte Carlo
simulations.
