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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 proceduresdoi:10.1214/15-ejs1026 fatcat:ffo2wdpbp5c6vbcsymuqa6ma6e