Introduction

Abstract

One may ask why Geometry, in particular differential geometry, is useful for statistics. The reason seems very simple and strong. A statistical model is a set of probability distributions to which we believe the true distribution belongs. It is a subset of all the possible probability distributions. In particular, a parametric model usually forms a finite-dimensional manifold imbedded in the set of all the possible probability distributions. In particular, a parametric model usually forms a finite-dimensional manifold imbedded in the set of all the possible probability distributions. For example a normal model consists of the probability distributions N(μ, σ2) parametrized by two parameters (μ, σ). The normal model M = {N(μ, σ2)} forms a two-dimensional manifold with coordinates μ and σ, and is imbedded in the set S = {p(x)} of all the regular probability distributions of a random variable x. One often uses a statistical model to carry out statistical inference, assuming that the true distribution is included in the model. However, a model is merely a hypothesis. The true distribution may not be in the model but be only close to it. Therefore, in order to evaluate statistical inference procedures, it is important to know what part the statistical model occupies in the entire set of probability distributions and what shape the statistical model has in the entire set.