  I define a natural measure of the complexity of a parametric distribution
relative to a given true distribution called the {\it razor} of a model family.
The Minimum Description Length principle (MDL) and Bayesian inference are shown
to give empirical approximations of the razor via an analysis that
significantly extends existing results on the asymptotics of Bayesian model
selection. I treat parametric families as manifolds embedded in the space of
distributions and derive a canonical metric and a measure on the parameter
manifold by appealing to the classical theory of hypothesis testing. I find
that the Fisher information is the natural measure of distance, and give a
novel justification for a choice of Jeffreys prior for Bayesian inference. The
results of this paper suggest corrections to MDL that can be important for
model selection with a small amount of data. These corrections are interpreted
as natural measures of the simplicity of a model family. I show that in a
certain sense the logarithm of the Bayesian posterior converges to the
logarithm of the {\it razor} of a model family as defined here. Close
connections with known results on density estimation and ``information
geometry'' are discussed as they arise.
