An Approximation of the Integrated Classification Likelihood for the Latent Block Model

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 block patterns are defined from the row and column classes. 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, sharing the same row and column classes. For a given data table, several candidate models are usually examined: they may differ in the numbers of clusters or more generally in the number of free parameters of the model. Model selection then becomes a critical issue, for which the tools that have been derived for model-based one-way 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 variant following a similar form. The proposed criteria are assessed on simulated data, where their performances are shown to be fairly reliable for medium to large data tables with well-separated clusters.