Group invariance for objective prior elicitation in testing problems

Authors: Luis Carvalho, Guido Consonni, Anabel Forte, Gonzalo García-Donato

In a Bayesian framework, Bayes factors and posterior probabilities of hypotheses are formal tools to approach testing problems with prior elicitation being one of the main challenges. This is a particularly delicate issue in an objective scenario where the prior information is scarce. In this context, Bayarri et al. (2012) introduced a number of criteria (also called `desiderata’) that objective priors should satisfy. Among these, the group invariance criterion emerged as a pioneering way of taking advantage of certain properties of the tested models to guide the construction of objective priors. In few words, the criterion states that if the entertained models share a group invariant structure this should be preserved after marginalization with respect to the prior. The criterion was applied by these authors in the Gaussian linear model, considering the basic location-scale group of invariance which is common to all linear models. The consequence was a (surprising) strong characterization of key aspects of conventional priors (like $g$-priors, and extensions) that didn’t have a formal justification until then. Nevertheless, other debatable aspects (like the popular use of the design matrix in the prior) remained open.

In this work (in progress) we explore the potential of the said criterion when a more specific group of invariance is applied to linear models with spherical-symmetric errors. Our main result is a refinement of the conditions on the priors found by Bayarri et al. (2012), ultimately leading to the conclusion that conventional priors are the only choice of priors that respect the group invariant properties in the posed problem. Almost accidentally, the question of the need of orthogonality among parameters enters into our development with potentially interesting conclusions for further related developments.