Consistency and Asymptotic Normality of Latent Blocks Model Estimators
Consistance et normalité asymptotique des estimateurs du modèle des blocs latents
Abstract
Latent Block Model (LBM) is a model-based method to cluster simultaneously the d columns and n rows of a data matrix. Parameter estimation in LBM is a difficult and multifaceted problem. Although various estimation strategies have been proposed and are now well understood empirically, theoretical guarantees about their asymptotic behavior is rather sparse. We show here that under some mild conditions on the parameter space, and in an asymptotic regime where log(d)/n and log(n)/d tend to 0 when n and d tend to +∞, (1) the maximum-likelihood estimate of the complete model (with known labels) is consistent and (2) the log-likelihood ratios are equivalent under the complete and observed (with unknown labels) models. This equivalence allows us to transfer the asymptotic consistency to the maximum likelihood estimate under the observed model. Moreover, the variational estimator is also consistent.
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