Information geometry

Summary

Information geometry studies families of probability distributions as manifolds, using the tools of differential geometry. The central structure is the Fisher information metric, a Riemannian metric on the parameter space, together with a family of affine connections called the $\alpha$-connections. On finite sample spaces, Chentsov's theorem shows that the Fisher metric is, up to a constant factor, the only metric invariant under sufficient statistics. The field was developed largely by Shun-ichi Amari and has applications in statistics, machine learning and information theory.