The semi-hierarchical Dirichlet Process and its application to clustering homogeneous distributions

TL;DR

This paper introduces the semi-hierarchical Dirichlet process, a new Bayesian prior for assessing homogeneity among multiple distributions, providing a more nuanced analysis than traditional binary tests.

Contribution

The paper proposes a semi-hierarchical Dirichlet process that extends existing models and addresses degeneracy issues, enabling detailed grouping of homogeneous populations.

Findings

The semi-hierarchical Dirichlet process effectively identifies homogeneous groups.

Theoretical properties of the model are established, including the behavior of the Bayes factor.

Simulation studies demonstrate the model's practical utility in real data applications.

Abstract

Assessing homogeneity of distributions is an old problem that has received considerable attention, especially in the nonparametric Bayesian literature. To this effect, we propose the semi-hierarchical Dirichlet process, a novel hierarchical prior that extends the hierarchical Dirichlet process of Teh et al. (2006) and that avoids the degeneracy issues of nested processes recently described by Camerlenghi et al. (2019a). We go beyond the simple yes/no answer to the homogeneity question and embed the proposed prior in a random partition model; this procedure allows us to give a more comprehensive response to the above question and in fact find groups of populations that are internally homogeneous when I greater or equal than 2 such populations are considered. We study theoretical properties of the semi-hierarchical Dirichlet process and of the Bayes factor for the homogeneity test when I = 2. Extensive simulation studies and applications to educational data are also discussed.