The role of algebraic methods and arguments in statistics and probability is
well-known, from Gauss' least squares principle through
A. Fisher's method of
variance decomposition. The work of A. James
according to E. Han-
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
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
M. Eaton
[Eat89], R.
The integral ofHaar, as didactically presented in L. Nachbin's
became a familiar tool among statisticians, and S. Anders-
son's group symmetry covariance models are recognized (e.g.,
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
on modelling complex stochastic systems us-
ing graph theory, and the monograph by G. Pistone, E. Riccomagno and H. Winn
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
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,
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