Estimation and Model Selection for Model-Based Clustering with the Conditional Classification Likelihood

TL;DR

This paper provides a theoretical analysis of the ICL criterion in model-based clustering, introducing the conditional classification likelihood and establishing its properties and relation to ICL.

Contribution

It introduces the conditional classification likelihood, studies its properties, and clarifies the theoretical relationship between ICL and this new criterion.

Findings

ICL is an approximation of the proposed criterion.

The new criterion has consistent estimation properties.

Insights into the class notion underlying ICL are provided.

Abstract

The Integrated Completed Likelihood (ICL) criterion has been proposed by Biernacki et al. (2000) in the model-based clustering framework to select a relevant number of classes and has been used by statisticians in various application areas. A theoretical study of this criterion is proposed. A contrast related to the clustering objective is introduced: the conditional classification likelihood. This yields an estimator and a model selection criteria class. The properties of these new procedures are studied and ICL is proved to be an approximation of one of these criteria. We oppose these results to the current leading point of view about ICL, that it would not be consistent. Moreover these results give insights into the class notion underlying ICL and feed a reflection on the class notion in clustering. General results on penalized minimum contrast criteria and on mixture models are derived, which are interesting in their own right.