  Using a recently proposed privacy definition of R\'enyi Differential Privacy
(RDP), we re-examine the inherent privacy of releasing a single sample from a
posterior distribution. We exploit the impact of the prior distribution in
mitigating the influence of individual data points. In particular, we focus on
sampling from an exponential family and specific generalized linear models,
such as logistic regression. We propose novel RDP mechanisms as well as
offering a new RDP analysis for an existing method in order to add value to the
RDP framework. Each method is capable of achieving arbitrary RDP privacy
guarantees, and we offer experimental results of their efficacy.
