Reference Priors for the Orbit in a Group Model

For a group model in which the group G acts freely on the parameter space Ω, this paper considers a prior which is a product of right Haar measure on G and a limiting form of Jeffreys' prior for the maximal invariant. When the parameter of interest is the orbit of G in Ω, it is shown that such a prior is the reference prior defined by Bernardo. A method of calculating this reference prior is given which avoids the necessity of working in a parameterization of Ω which expresses Ω as a product of G and a cross section. Examples of the multivariate normal distribution, with the parameter of interest being the correlation matrix or the eigenvalues of the covariance matrix, are discussed.