Integrated Classification Likelihood for Model selection in Block Clustering

Block clustering (or co-clustering or simultaneous clustering) aims at simultaneously partitioning the rows and columns of a data table to reveal homogeneous block structures. This structure can stem from the latent block model which provides a probabilistic modelling of data tables whose blocks arise from row and column clusters. For continuous data, each table entry is typically assumed to follow a Gaussian distribution whose parameters are common to all entries belonging to the same block, that is, with identical row and column classes. Several candidate models can be adjusted to a given data table: they may differ in the numbers of clusters or more generally in the number of free parameters. Model selection then becomes a critical issue, for which the tools that have been derived for model-based oneway clustering need to be adapted. We develop here a criterion based on an approximation of the integrated classification likelihood (ICL) of block models, and propose a BIC-like criterion derived from the form obtained. The proposed criteria are illustrated by experiments on simulated data, where their performances are shown to be best reliable for medium to large data tables with well-separated clusters.