Privacy Guarantees in Posterior Sampling under Contamination

In recent years, differential privacy has been adopted by tech-companies and governmental agencies as the standard for measuring privacy in algorithms. In this article, we study differential privacy in Bayesian posterior sampling settings. We begin by considering differential privacy in the most common privatisation setting in which Laplace or Gaussian noise is injected into the output. In an effort to achieve better differential privacy, we consider adopting {\em Huber's contamination model} for use within privacy settings, and replace at random data points with samples from a heavy-tailed distribution ({\em instead} of injecting noise into the output). We derive bounds for the differential privacy level $(\epsilon,\delta)$ of our approach, without requiring bounded observation and parameter spaces, a restriction commonly imposed in the literature. We further consider for our approach the effect of sample size on the privacy level and the rate at which $(\epsilon,\delta)$ converges to zero. Asymptotically, our contamination approach is fully private with no information loss. We also provide examples of inference models for which our approach applies, with theoretical convergence rate analysis and simulation studies.