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The Integrated Completed Likelihood (ICL) criterion has been proposed byBiernacki et al. (2000) in the model-based clustering framework to select arelevant number of classes and has been used by statisticians in variousapplication areas. A theoretical study of this criterion is proposed. Acontrast related to the clustering objective is introduced: the conditionalclassification likelihood. This yields an estimator and a model selectioncriteria class. The properties of these new procedures are studied and ICL isproved to be an approximation of one of these criteria. We oppose these resultsto the current leading point of view about ICL, that it would not beconsistent. Moreover these results give insights into the class notionunderlying ICL and feed a reflection on the class notion in clustering. Generalresults on penalized minimum contrast criteria and on mixture models arederived, which are interesting in their own right.

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