The Integrated Completed Likelihood (ICL) criterion was introduced by Biernacki, Celeux and Govaert (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 ICL is proposed. A contrast related to the clustering objective is introduced: the conditional classification likelihood . An estimator and model selection criteria are deduced. The properties of these new procedures are studied and ICL is proved to be an approximation of one of these criteria. We contrast these results with 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 upper-bounds of the bracketing entropy in parametric situations are derived, which can be useful per se. Practical solutions for the computation of the introduced procedures are proposed, notably an adapted EM algorithm and a new initialization method for EM-like algorithms which helps to improve the estimation in Gaussian mixture models.
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