  Model complexity plays an essential role in its selection, namely, by
choosing a model that fits the data and is also succinct. Two-part codes and
the minimum description length have been successful in delivering procedures to
single out the best models, avoiding overfitting. In this work, we pursue this
approach and complement it by performing further assumptions in the parameter
space. Concretely, we assume that the parameter space is a smooth manifold, and
by using tools of Riemannian geometry, we derive a sharper expression than the
standard one given by the stochastic complexity, where the scalar curvature of
the Fisher information metric plays a dominant role. Furthermore, we derive the
minmax regret for general statistical manifolds and apply our results to derive
optimal dimensional reduction in the context of principal component analysis.
