  We use information geometry, in which the local distance between models
measures their distinguishability from data, to quantify the flow of
information under the renormalization group. We show that information about
relevant parameters is preserved, with distances along relevant directions
maintained under flow. By contrast, irrelevant parameters become less
distinguishable under the flow, with distances along irrelevant directions
contracting according to renormalization group exponents. We develop a
covariant formalism to understand the contraction of the model manifold. We
then apply our tools to understand the emergence of the diffusion equation and
more general statistical systems described by a free energy. Our results give
an information-theoretic justification of universality in terms of the flow of
the model manifold under coarse-graining.
