Maximum likelihood estimation for the Dirichlet distribution
Sucharitha Dodamgodage, Thevasha Sathiyakumar, Daniel T. Fuller, Shantanu Sur, Sumona Mondal, Nabendu Pal · Communication in Statistics- Theory and Methods · 2025
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, a significant 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 analytically 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 (Citation1989). In the course of our investigation, we have also proved a conjecture left open by Ronning (Citation1989) for the computation of the MLE, thereby bringing closure to Ronning’s (Citation1989) work. Next, we consider a symmetric Dirichlet distribution, a restricted family of the general Dirichlet model that is encountered in a hypothesis testing problem, where the null hypothesis states that all components of the random vector following the Dirichlet model are equally represented, on average. We again prove analytically the existence and uniqueness of the MLE for a single scalar parameter of the symmetric Dirichlet distribution. The proof of the latter result is more complex than the derivation of the MLE in the general non symmetric case and, to the best of our knowledge, does not appear in the existing literature.