A Closed-Form Approximation to the Conjugate Prior of the Dirichlet and Beta Distributions
This provides a practical solution for Bayesian statisticians and machine learning practitioners working with Dirichlet and beta likelihoods, though it is incremental as it builds on known conjugate prior theory.
The paper tackles the intractability of the conjugate prior for Dirichlet and beta distributions by deriving a closed-form approximation, enabling fully tractable Bayesian conjugate treatment without Monte Carlo simulations.
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.