Editing Haar measure (section)

===Mathematical statistics===
In mathematical statistics, Haar measures are used for prior measures, which are [[prior probability|prior probabilities]] for compact groups of transformations. These prior measures are used to construct [[admissible procedure]]s, by appeal to the characterization of admissible procedures as [[Bayesian statistics|Bayesian procedures]] (or limits of Bayesian procedures) by [[Abraham Wald|Wald]]. For example, a right Haar measure for a family of distributions with a [[location parameter]] results in the [[Pitman Estimator|Pitman estimator]], which is [[minimum variance unbiased estimator|best]] [[equivariant estimator|equivariant]]. When left and right Haar measures differ, the right measure is usually preferred as a prior distribution. For the group of affine transformations on the parameter space of the normal distribution, the right Haar measure is the [[Jeffreys prior]] measure<!-- improper "prior", not a probability distribution  -->.<ref>{{Citation | first = James O.| last = Berger | author-link = James O. Berger | title = Statistical decision theory and Bayesian analysis | year = 1985 | edition=second|chapter=6 Invariance|pages = 388–432 |publisher=Springer Verlag| bibcode = 1985sdtb.book.....B }}</ref> Unfortunately, even right Haar measures sometimes result in useless priors, which cannot be recommended for practical use, like other methods of constructing prior measures that avoid subjective information.<ref>{{Cite book|author=Robert, Christian P|title=The Bayesian Choice – A Decision-Theoretic Motivation|publisher=Springer|year=2001|edition=second|isbn=0-387-94296-3}}</ref>

Another use of Haar measure in statistics is in [[conditional inference]], in which the sampling distribution of a statistic is conditioned on another statistic of the data. In invariant-theoretic conditional inference, the sampling distribution is conditioned on an invariant of the group of transformations (with respect to which the Haar measure is defined). The result of conditioning sometimes depends on the order in which invariants are used and on the choice of a [[maximal invariant]], so that by itself a  [[statistical principle]] of invariance fails to select any unique best conditional statistic (if any exist); at least another principle is needed.

For non-compact groups, statisticians have extended Haar-measure results using [[amenable group]]s.<ref>{{cite journal | last1 = Bondar | first1 = James V. | last2 = Milnes | first2 = Paul | year = 1981 | title = Amenability: A survey for statistical applications of Hunt–Stein and related conditions on groups | journal = Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete | volume = 57 | issue = 1 | pages = 103–128 | doi = 10.1007/BF00533716 | doi-access = free }}</ref>