Preface

The role of algebraic methods and arguments in statistics and probability is

well-known, from Gauss' least squares principle through

R.

A. Fisher's method of

variance decomposition. The work of A. James

[Jam57],

according to E. Han-

nan

[Han65,

p.65], appears to be among the first describing the group-theoretic

nature of Fisher's argument, giving meaning to the notion of

relationship algebra,

or the commuting algebra of a representation of the group of symmetries of an

experimental design. During the same period, U. Grenander

[Gre63]

showed the

effectiveness of harmonic analysis techniques to extend the classical limit theorems

to algebraic structures such as locally compact groups, Banach spaces and topo-

logical algebras. Two decades later, the relevance of group invariance and group

representation arguments in statistical inference would become evident in the works

of, among others, P. Diaconis

[Dia88],

M. Eaton

[Eat89], R.

Farrell

[Far85]

and

R.

Wijsman

[Wij90].

The integral ofHaar, as didactically presented in L. Nachbin's

monograph

[Nac65],

became a familiar tool among statisticians, and S. Anders-

son's group symmetry covariance models are recognized (e.g.,

[Per87])

now as a

landmark concept extending and setting definitive boundaries to the standard of

multivariate statistical analysis, as known in the tradition of T.W. Anderson. The

contributions of S. Lauritzen

[Lau96]

on modelling complex stochastic systems us-

ing graph theory, and the monograph by G. Pistone, E. Riccomagno and H. Winn

[PRWOO]

on computational commutative algebra in statistics are examples of re-

cent new complements to the seminal work of Alan James.

The opportunity present to all of us statisticians and probabilists today is in

the formulation and application of comprehensive concepts and language capable

of capturing in their essence the research questions posed by those with whom we

collaborate - reasoning tools which represent the broad physical, biological, behav-

ioral, social interface between phenomena and data. The challenge is in representing

and including the interface, the context, in our proposed methods of scientific expla-

nation. As recently observed by V. Lakshminarayanan

[LSJ98],

group-theoretical

analysis may lead to prediction of new measurable effects or allow newer methods

of analysis of data and offer greater insight into [specific] research hypotheses. This

remark exemplifies the many opportunities ahead.

The present volume was originally proposed by the editors during the AMS

Sectional Meeting, in Montreal, 1997. A later sectional meeting in South Bend,

Indiana, in the Spring of 2000, gave us the opportunity to continue with the project.

The papers appearing in the

South Bend

volume result from presented talks and

subsequent invited contributions.

We are grateful to all who assisted us during the refereeing process, namely

J. Aitchison, S. Andersson, J-F. Burnol, P. Chan, T. Chang, D. Collombier,

ix