Revisiting Dirichlet Mixture Model: unraveling deeper insights and practical applications

This study revisits the Dirichlet Mixture Model (DMM), offering comprehensive insights into specific facets of parameter estimation. Estimating parameters of the DMM is challenging, with previous approaches focusing on standard parametrization, which lacks interpretability. We propose an alternative parametrization of the Dirichlet distribution using mean and precision, which provides critical insights into the distribution’s location and peakedness. This parametrization is versatile, covering a wide range of scenarios with varying locations and precision levels, making it applicable to diverse datasets. Depending on whether one or both parameters are unknown, the estimation procedure varies, and estimates also differ when precision is identical across mixture components. In this article, we introduce this alternative parametrization and meticulously explore four distinct scenarios, deriving maximum likelihood estimates (MLE) for each using the Expectation-Maximization (EM) algorithm. For high-dimensional data, where standard methods often falter due to additional challenges, we present an innovative estimation approach utilizing Stirling’s approximation and moment approximation, which provides closed-form solutions and faster execution times. Our study demonstrates the identifiability of the DMM and employs a closed-form approximation for Kullback–Leibler (KL) divergence to evaluate goodness of fit. Practical applications are illustrated through the analysis of both simulated and real datasets, showcasing the practical utility of the DMM.