Model selection for the binary latent block model

The latent block model is a mixture model that can be used to deal with the simultaneous clustering of rows and columns of an observed numerical matrix, known as co-clustering. For this mixture model unfortunately, neither the likelihood, nor the EM algorithm are numerically tractable, due to the dependence of the rows and columns into the label joint distribution conditionally to the observations. Several approaches can be considered to compute approximated solutions, for the maximum likelihood estimator as well as for the likelihood itself. The comparison of a determinist approach using a variational principle with a stochastic approach using a MCMC algorithm is first discussed and applied in the context of binary data. These results are then used to build and compute ICL and BIC criteria for model selection. Numerical experiments show the interest of this approach in model selection and data reduction.