We explore the consequences of adjoining a symmetry group to a statistical model. Group actions are first
induced on the sample space, and then on the parameter space. It is argued that the right invariant measure
induced by the group on the parameter space is a natural non-informative prior for the parameters
of the model. The permissible sub-parameters are introduced, i.e., the subparameters upon which
group actions can be defined. Equivariant estimators are similarly defined. Orbits of the group
are defined on the sample space and on the parameter space; in particular the group action is
called transitive when there is only one orbit. Credibility sets and confidence sets are shown
(under right invariant prior and assuming transitivity on the parameter space) to be equal when
defined by permissible sub-parameters and constructed from equivariant estimators. The effect of
different choices of transformation group is illustrated by examples, and properties of the orbits
on the sample space and on the parameter space are discussed. It is argued that model reduction should
be constrained to one or several orbits of the group. Using this and other natural criteria and concepts,
among them concepts related to design of experiments under symmetry, leads to links towards chemometrical
prediction methods and towards the foundation of quantum theory.
@article{1102516480,
author = {Helland, Inge S.},
title = {Statistical Inference under Symmetry},
journal = {Internat. Statist. Rev.},
volume = {72},
number = {1},
year = {2004},
pages = { 409-422},
language = {en},
url = {http://dml.mathdoc.fr/item/1102516480}
}
Helland, Inge S. Statistical Inference under Symmetry. Internat. Statist. Rev., Tome 72 (2004) no. 1, pp. 409-422. http://gdmltest.u-ga.fr/item/1102516480/