Invariant Conjugate Analysis for Exponential Families

Abstract

There are several ways to parameterize a distribution belonging to an exponential family, each one leading to a different Bayesian analysis of the data under standard conjugate priors. To overcome this problem, we propose a new class of conjugate priors which is invariant with respect to smooth reparameterization. This class of priors contains the Jeffreys prior as a special case, according to the value of the hyperparameters. Moreover, these conjugate distributions coincide with the posterior distributions resulting from a Jeffreys prior. Then these priors appear naturally when several datasets are analyzed sequentially and when the Jeffreys prior is chosen for the first dataset. We apply our approach to inverse Gaussian models and propose full invariant analyses of three datasets.