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We derive the conjugate prior of the Dirichlet and beta distributions andexplore it with numerical examples to gain an intuitive understanding of thedistribution itself, its hyperparameters, and conditions concerning itsconvergence. Due to the prior's intractability, we proceed to define andanalyze a closed-form approximation. Finally, we provide an algorithmimplementing this approximation that enables fully tractable Bayesian conjugatetreatment of Dirichlet and beta likelihoods without the need for Monte Carlosimulations.

A Closed-Form Approximation to the Conjugate Prior of the Dirichlet and Beta Distributions