Sélection de modèles: de la théorie à la pratique
Journal de la Société Française de Statistique
Since the seminal work of Akaike in the early seventies, optimizing some penalized empirical criterion such as the penalized log-likelihood has become a classical solution to the problem of choosing a proper statistical model from the data.
For many model selection problems such as multiple change-point detection and variable selection for instance, it is desirable to let the dimension or the number of models of a given dimension grow with the sample size. A non asymptotic theory for model selection has therefore emerged during these last ten years in order to take this type of situations into account. The main issue both from a practical and a theoretical view point is to understand how to penalize an empirical criterion such as the log-likelihood in order to get some optimal selection procedure. Asymptotic theory provides some useful indications on the shape of the penalty but it often
leaves to the user the choice of numerical constants. The optimal value for these constants is generally unknown. In some situations theory is indeed not sharp enough to lead to explicit values. In some other cases, the problem is more of a statistical nature since according to the theory, the optimal value should depend on the unknown distribution of the observations. Our purpose here is to promote some data-driven method to calibrate the penalty. This method is partly based
on preliminary theoretical results that we shall recall and partly founded on some heuristics that we intend to explain.