Maximum likelihood estimation for the Dirichlet distribution

The Dirichlet distribution is a multivariate generalization of the Beta distribution, defining a family of unit sum-constrained probabilities or proportions in a multi-dimensional simplex. This distribution is usually the first choice in modeling compositional data and has been applied in various fields, including modeling microbiome data, text classification, and market share analysis. The existing literature suggests that the maximum likelihood estimator (MLE) is the most effective method for estimating Dirichlet parameters. However, asignificant issue is that simply assuming the existence and uniqueness of the MLE for the Dirichlet model parameter without an analytic proof can lead to a meaningless interpretation of its bias and/or relative mean squared error. First, we address this problem by proving the existence and uniqueness of MLEs for the general Dirichlet distribution parameters. Our method relies on a particular representation of the digamma function, and our proof is much simpler than the one by Ronning (1989). In the course of our investigation, we have also proved a conjecture left open by Ronning (1989) for the computation of the MLE, thereby bringing

Apery’s constant

Beta function

Digamma function

Euler’s constant

Log-likelihood function

Trigamma function 

IMS