A unified approach to marginal equivalence in the general framework of group invariance

Two Bayesian models with different sampling densities are said to be marginally equivalent if the joint distribution of observables and the parameter of interest is the same for both models. We discuss marginal equivalence in the general framework of group invariance. We introduce a class of sampling models and establish marginal equivalence when the prior for the nuisance parameter is relatively invariant. We also obtain some robustness properties of invariant statistics under our sampling models. Besides the prototypical example of $v$-spherical distributions, we apply our general results to two examples---analysis of affine shapes and principal component analysis.