Hierarchical Mixture of Finite Mixtures

TL;DR

This paper introduces a hierarchical mixture model based on novel Bayesian priors for multilevel data, offering improved analytical and computational capabilities over existing methods like the Hierarchical Dirichlet Process.

Contribution

It extends the Mixture of Finite Mixture model to a hierarchical framework with a full distribution theory and efficient inference algorithms, enhancing clustering and analysis of multilevel data.

Findings

Outperforms Hierarchical Dirichlet Process in clustering accuracy and computational efficiency.

Successfully applied to shot put data to identify athlete performance patterns across seasons.

Provides a flexible Bayesian framework for multilevel data analysis.

Abstract

Statistical modelling in the presence of data organized in groups is a crucial task in Bayesian statistics. The present paper conceives a mixture model based on a novel family of Bayesian priors designed for multilevel data and obtained by normalizing a finite point process. In particular, the work extends the popular Mixture of Finite Mixture model to the hierarchical framework to capture heterogeneity within and between groups. A full distribution theory for this new family and the induced clustering is developed, including the marginal, posterior, and predictive distributions. Efficient marginal and conditional Gibbs samplers are designed to provide posterior inference. The proposed mixture model overcomes the Hierarchical Dirichlet Process, the utmost tool for handling multilevel data, in terms of analytical feasibility, clustering discovery, and computational time. The motivating application comes from the analysis of shot put data, which contains performance measurements of athletes across different seasons. In this setting, the proposed model is exploited to induce clustering of the observations across seasons and athletes. By linking clusters across seasons, similarities and differences in athletes' performances are identified.