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

TL;DR

This paper derives the conjugate prior for Dirichlet and beta distributions, introduces a closed-form approximation to address intractability, and provides an algorithm for fully tractable Bayesian inference without Monte Carlo methods.

Contribution

It presents the first closed-form approximation to the conjugate prior of Dirichlet and beta distributions, enabling efficient Bayesian analysis.

Findings

The approximation closely matches the true conjugate prior in numerical tests.

The algorithm allows for tractable Bayesian inference without Monte Carlo simulations.

The approach improves computational efficiency in Bayesian modeling with Dirichlet and beta distributions.

Abstract

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.