Asymptotic law of likelihood ratio for multilayer perceptron models.

Joseph Rynkiewicz (1)

1 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)

We consider regression models involving multilayer perceptrons (MLP) with one hidden layer and a Gaussian noise. The data are assumed to be generated by a true MLP model and the estimation of the parameters of the MLP is done by maximizing the likelihood of the model. When the number of hidden units of the true model is known, the asymptotic distribution of the maximum likelihood estimator (MLE) and the likelihood ratio (LR) statistic is easy to compute and converge to a $\chi^2$ law. However, if the number of hidden unit is over-estimated the Fischer information matrix of the model is singular and the asymptotic behavior of the MLE is unknown. This paper deals with this case, and gives the exact asymptotic law of the LR statistics. Namely, if the parameters of the MLP lie in a suitable compact set, we show that the LR statistics is the supremum of the square of a Gaussian process indexed by a class of limit score functions.

Mots clés: Multilayer perceptron; asymptotic statistics; model selection; Gaussian processes

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Joseph Rynkiewicz. Asymptotic law of likelihood ratio for multilayer perceptron models.. 2010. ⟨hal-00540383⟩