  This paper introduces a new notion of dimensionality of probabilistic models
from an information-theoretic view point. We call it the "descriptive
dimension"(Ddim). We show that Ddim coincides with the number of independent
parameters for the parametric class, and can further be extended to real-valued
dimensionality when a number of models are mixed. The paper then derives the
rate of convergence of the MDL (Minimum Description Length) learning algorithm
which outputs a normalized maximum likelihood (NML) distribution with model of
the shortest NML codelength. The paper proves that the rate is governed by
Ddim. The paper also derives error probabilities of the MDL-based test for
multiple model change detection. It proves that they are also governed by Ddim.
Through the analysis, we demonstrate that Ddim is an intrinsic quantity which
characterizes the performance of the MDL-based learning and change detection.
