Invariance, Haar Measures, and Equivariant Estimators
Invariance can be seen as a notion introduced in frequentist settings to restrict the range of acceptable estimators sufficiently so that an optimal estimator can be derived. From this point of view, it appears as an alternative to unbiasedness and is thus similarly at odds with the Bayesian paradigm. However, it also partakes from a non-decision-theoretic heuristic, namely that estimators should meet some consistency requirements under a group of transformations, and it is thus of interest to consider this notion. Moreover, a Bayesian perspective of invariance is justified by the fact that optimal (equivariant) estimators are always Bayes or generalized Bayes estimators. The corresponding measures can be considered as noninformative priors induced by the invariance structure. Therefore, a Bayesian study of invariance is appealling, not because classical optimality once more relies on Bayesian estimators, but mainly because of the connection between invariance structures and the derivation of noninformative distributions.
KeywordsMaximum Likelihood Estimator Location Parameter Haar Measure Invariance Structure Quadratic Loss
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