Differential Geometry and Statistics PDF

Preface

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 cts 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.

The second chapter, in which we introduce manifolds, should be most useful to statisticians who want to learn about the subject. The traditional theory starts with a heavy meal of the purest mathematics, (topological spaces, co-ordinate coverings, differentiable functions), before embarking on a treatment of calculus that is filled with multilinear algebra, and bears little relationship to anything one might have learned about several-variable calculus as an undergraduate. By contrast our treatment starts with calculus on manifolds as a geometrical approach to the theory of rates of change of functions, treating it as though it were a first course on several variable calculus. We explain how the several-variable chain rule can be interpreted as dividing variations through a point into families with different velocities, how df is to be interpreted as the rate of change of / as a function of velocity, and what are vector fields (contravariant 1-tensors) and 1-forms (covariant 1- tensors). We give a brief discussion of the foundational concepts of differentiability and manifolds at the end of the chapter, but these are not really important for the application of differential geometry to statistics.

Our comment on the great divide between the so-called coordinate- free and index-laden approaches to differential geometry, is that we aim to be geometrical without being obsessed with freedom from co-ordinates. We have enormous interest in co-ordinates when it comes to calculations. However, it seems pointless to us to be in the position either to be able to calculate everything but explain nothing, or to explain everything but calculate nothing. So we explain geometrical concepts in co-ordinate-free terms, and we translate them into co-ordinate systems for calculations, with whatever debauches of indices they require

Once the basic notions are in place, most notably the definition in Chapter 2 of the tangent space to a manifold, we begin an elaboration of the parts of differential geometry that are useful in statistics, illustrating them with statistical applications and examples. As the number of statistical applications is growing rapidly we have been unable to consider them all. However we believe that we have covered all the concepts from differential geometry that are needed at this point in time. Chapter 3 explains the idea of submanifold and the definition of a statistical manifold. We mention again the simplest statistical manifolds, the exponential families, and then consider the families with a high degree of symmetry, the transformation models.

The next two chapters introduce the concept of connections and their curvature, Amari’s a-connections and the theory of statistical divergences. A connection defines the rate of change of vector fields. It therefore tells us which curves have constant tangent vector fields, that is which curves are straight lines or geodesics. Hence a connection defines a notion of geometry, or straight lines and the different connections define different geometries. Some connections are essentially 'flat'. That is, the geometry they define is Euclidean. The curvature of a connection is a measure of its departure from flatness

Chapter 7 introduces the maximum likelihood estimator and considers some results in asymptotics, in particular the work of Amari. Here we begin to see the importance of Taylor series and the need for a higher-order geometry in statistics. The final Chapters 8 and 9 consider this higher order geometry: the theory of strings or phyla developed by Barndorff-Neilsen and Blaesild. Strings are generalisations of tensors. If we think of tensors in co-ordinates as functions with many indices transforming under a change of coordinates by the first derivative of the co-ordinate transformation, then a string has more indices and transforms by higher derivatives of the co-ordinate transformation. To consider strings from a coordinate- free point of view requires that we introduce in Chapter 8 the theory of principal and vector bundles, in particular the socalled infinite frame bundle and the infinite phylon group. Chapter 9 then applies this theory to Taylor expansions and co-ordinate strings and relates the theory of strings to the representation theory of the infinite phylon group.

A book is not just the result of the labours of its authors but also of the generosity of others. First and foremost we thank our families who had to live through this book’s production; then our many statistical colleagues who have laboured to explain their subject to us. Our thanks and apologies for the [)laces where despite your efforts we get it wrong. Particular thanks must go to Peter McCullagh for providing us with T^X macros for this book and to Peter Jupp for his amazingly thorough reading of our first manuscript. Of course any remaining errors and omissions are our responsibility.

Author: M.K. Murray, J.W. Rice