Reference Priors for the Orbit in a Group Model

Abstract

For a group model in which the group $\mathbf{G}$ acts freely on the parameter space $\mathbf{\Omega}$, this paper considers a prior which is a product of right Haar measure on $\mathbf{G}$ and a limiting form of Jeffreys' prior for the maximal invariant. When the parameter of interest is the orbit of $\mathbf{G}$ in $\mathbf{\Omega}$, 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 $\mathbf{\Omega}$ which expresses $\mathbf{\Omega}$ as a product of $\mathbf{G}$ 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.