Group invariant inferred distributions via noncommutative probability

Abstract

One may consider three types of statistical inference: Bayesian, frequentist, and group invariance-based. The focus here is on the last method. We consider the Poisson and binomial distributions in detail to illustrate a group invariance method for constructing inferred distributions on parameter spaces given observed results. These inferred distributions are obtained without using Bayes' method and in particular without using a joint distribution of random variable and parameter. In the Poisson and binomial cases, the final formulas for inferred distributions coincide with the formulas for Bayes posteriors with uniform priors.Comment: Published at http://dx.doi.org/10.1214/074921706000000563 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org