Monte Carlo Methods For Data Privacy Applications

This thesis focuses on data privacy applications with Bayesian inference, particularly Markov chain Monte Carlo (MCMC) methods for two main data privacy problems. Firstly, we focus on statistic selection with a Fisher information, and we show that informativeness and efficiency are closely related in the differential privacy setting. Then, we propose a novel generative model for the private linear regression that outshines state-of-art methods. In this work, MCMC algorithms are specifically developed for several data privacy settings. While some of the settings enable to work with simple and efficient Metropolis-Hastings (MH), others require more advanced sampling methods such as Pseudo-Marginal Metropolis-Hastings (PMMH), Metropolis-Hastings with Averaged Acceptance Ratios (MHAAR) or MH-within-Gibbs sampling. In detail, we prefer using versions of MH, PMMH, MHAAR for the statistic selection, and derivatives of the MH-within-Gibbs for the linear regression problem. At the end, we conduct several numerical experiments for evaluation purposes. In the statistic selection part, we rigorously deal with each problem setting and we obtain that Fisher information is actually a useful tool for the differential privacy applications for almost all possible problem definitions. For the linear regression, both simulated and real datasets are tested, and we observe that proposed methods beat existing algorithms in terms of efficiency and effectiveness.