Differential Geometry and Statistics

M.K. Murray, J.W. Rice

• Publisher: Routledge
• Year: 2017
• Print ISBN:
  9780412398605
• Edition: 1
• eBook ISBN:
  9781351455114
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Several years ago our statistical friends and relations introduced us to the work of Amari and Barndorff-Nielsen on applications of differential geometry to statistics. This book has arisen because we believe that there is a deep relationship between statistics and differential geometry and moreoever that this relationship uses parts of differential geometry, particularly its 'higher-order' aspects not readily accessible to a statistical audience from the existing literature. It is, in part, a long reply to the frequent requests we have had for references on differential geometry! While we have not gone beyond the path-breaking work of Amari and Barndorff- Nielsen in the realm of applications, our book gives some new explanations of their ideas from a first principles point of view as far as geometry is concerned. In particular it seeks to explain why geometry should enter into parametric statistics, and how the theory of asymptotic expansions involves a form of higher-order differential geometry. The first chapter of the book explores exponential families as flat geometries. Indeed the whole notion of using log-likelihoods amounts to exploiting a particular form of flat space known as an affine geometry, in which straight lines and planes make sense, but lengths and angles are absent. We use these geometric ideas to introduce the notion of the second fundamental form of a family whose vanishing characterises precisely the exponential families.

CHAPTER 1

The geometry of exponential families

1.1 Geometry, parameters and co-ordinates

Parametric statistics concerns parametrised families of probability distributions p(θ), where the parameter θ = (θ¹,…, θ^d) varies over some open set in R^d. The most common example is the normal family, which is usually expressed as a family of densities

$p(\mu ,\sigma )=N(\mu ,{\sigma }^2)=\frac{1}{\sqrt{2\pi \sigma }}\exp (\frac{-{(x-\mu )}^2}{2{\sigma }^2})$

The parameter θ in this case is the pair (μ, σ) which varies over the open subset of R² determined by μ > 0. The sample space is R and the densities are with respect to Lebesgue measure dx on R, so that as a set of probability measures the normal family is

$\mathcal{N}=\left\{p(\mu ,\sigma )dx|\mu \in R,\sigma >0\right\}$

Statistical inference concerns situations in which one knows or suspects that data are generated by sampling from a space according to a probability distribution which is a member of some known family p(θ). The problem is to infer facts about the distribution from the data. For example, one might want to know the parameter value of the distribution (point estimation), or simply whether or not this value lies in some particular set of parameters (hypothesis testing). If a given collection of numbers has arisen by sampling from a normal distribution then one might ask which normal distribution it is, i.e. what are the values of μ and σ for this particular normal distribution. On the other hand one might only want to test the hypothesis that the mean is greater than 1.

Many of these tests and much of the theory of statistical inference depends on the choice of parameters. This dependence on parameters usually comes about because the theory applies differential calculus to these parameters; differentiating them or perhaps Taylor series expanding some function with respect to them. It is important to know how the theory depends on the parameters, either because one suspects that it should not depend on the parameters at all or because one would like to know if a particular choice of parameters may simplify matters. That part of differential geometry which we have called ‘Calculus on Manifolds’ in Chapter 2, is concerned exactly with this question of how the differential calculus depends on co-ordi...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• 1. The geometry of exponential families
• 2. Calculus on manifolds
• 3. Statistical manifolds
• 4. Connections
• 5. Curvature
• 6. Information metrics and statistical divergences
• 7. Asymptotics
• 8. Bundles and tensors
• 9. Higher order geometry
• References
• Notation index
• Subject index