Full Robustness in Bayesian Modelling of a Scale Parameter

Abstract

Conflicting information, arising from prior misspecification or outlying observations, may contaminate the posterior inference in Bayesian modelling. The use of densities with sufficiently heavy tails usually leads to robust posterior inference, as the influence of the conflicting information decreases with the importance of the conflict. In this paper, we study full robustness in Bayesian modelling of a scale parameter. The log-slowly, log-regularly and log-exponentially varying functions as well as log-exponential credence (LE-credence) are introduced in order to characterize the tail behaviour of a density. The asymptotic behaviour of the marginal and the posterior is described and we find that the scale parameter given the complete information converges in distribution to the scale given the non-conflicting information, as the conflicting values (outliers and/or prior’s scale) tend to 0 or $\infty$, at any given rate. We propose a new family of densities defined on $\mathbb{R}$ with a large spectrum of tail behaviours, called generalized exponential power of the second form (GEP₂), and its exponential transformation defined on $(0,\infty )$, called log-GEP₂, which proves to be helpful for robust modelling. Practical considerations are addressed through a case of combination of experts’ opinions, where non-robust and robust models are compared.