Bayesian marginal equivalence of elliptical regression models

Abstract

The use of proper prior densities in regression models with multivariate non-Normal elliptical error distributions is examined when the scale matrix is known up to a precision factor τ, treated as a nuisance parameter. Marginally equivalent models preserve the convenient predictive and posterior results on the parameter of interest β obtained in the reference case of the Normal model and its conditionally natural conjugate gamma prior. Prior densities inducing this property are derived for two special cases of non-Normal elliptical densities representing very different patterns of tail behaviour. In a linear framework, so-called semiconjugate prior structures are defined as leading to marginal equivalence to a Normal data density with a fully natural conjugate prior.