Clustering methods often come down to the optimization of a numeric criterion defined from a distance or from a dissimilarity measure. It is possible to show that this problem is often equivalent to the estimation of the parameters of a probabilistic model under the classification likelihood approach. For instance, we know that the inertia criterion optimized under the k-means algorithm corresponds to the hypothesis of a population arising from a Gaussian mixture. In this paper, we propose an adapted mixture model for categorical data. Using the classification likelihood approach, we develop the Classification EM algorithm (CEM) to estimate the parameters of the mixture model. With our probabilistic model, the data are not denatured and the estimated parameters readily indicate the characteristics of the clusters. This probabilistic approach gives an interpretation of the criterion optimized by the k-modes algorithm which is an extension of k-means to categorical attributes and allows us to study the behavior of this algorithm.
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