This paper considers the use of local $phi$-divergence measures between posterior distributions under classes of perturbations in order to investigate the inherent robustness of certain classes. The smaller value of the limiting local $phi$-divergence implies more robustness for the prior or the likelihood. We consider the cases when the likelihood comes form the class of weighted distribution. Two kinds of perturbations are considered for the local sensitivity analysis. In addition, some numerical examples are considered which provide measures of robustness.