Testing Homogeneity for Normal Mixture Models: Variational Bayes Approach

TL;DR

This paper introduces a novel variational Bayes-based method for testing homogeneity in normal mixture models, addressing the challenge of singular parameter spaces and providing a new theoretical foundation validated by numerical experiments.

Contribution

It develops a new hypothesis testing framework for normal mixture homogeneity using variational Bayes, clarifying the asymptotic behavior of the free energy's constant term.

Findings

Theoretical analysis of the free energy's asymptotic behavior.

Validation of the proposed method through numerical experiments.

Improved understanding of homogeneity testing in mixture models.

Abstract

The test of homogeneity for normal mixtures has been conducted in diverse research areas, but constructing a theory of the test of homogeneity is challenging because the parameter set for the null hypothesis corresponds to singular points in the parameter space. In this paper, we examine this problem from a new perspective and offer a theory of hypothesis testing for homogeneity based on a variational Bayes framework. In the conventional theory, the constant order term of the free energy has remained unknown, however, we clarify its asymptotic behavior because it is necessary for constructing a hypothesis test. Numerical experiments shows the validity of our theoretical results.