  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.
