Strong Asymptotic Assertions for Discrete MDL in Regression and Classification

We study the properties of the MDL (or maximum penalized complexity) estimator for Regression and Classification, where the underlying model class is countable. We show in particular a finite bound on the Hellinger losses under the only assumption that there is a “true” model contained in the class. This implies almost sure convergence of the predictive distribution to the true one at a fast rate. It corresponds to Solomono’s central theorem of universal induction, however with a bound that is exponentially larger.