A Bayesian analysis of the linear regression model with only parts of the prior distribution specified or a robust Bayesian analysis lead to sets of posterior distributions. The so-called feasible ellipsoid of Leamer (1978) describes the region of posterior means if for conjugate priors the prior covariance matrix is varying in the set of symmetric positive-definite matrices. As an extension to Bayesian confidence sets (HPD regions, i.e. regions of highest posterior probability) we introduce the concept of HiFi (high fiduciary) regions. The HiFi region is a union of HPD regions, and is a tool for describing the dependence of the posterior distribution on the prior covariance. In this paper we assume that the prior covariance matrix varies in an interval of positive definite matrices.