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Abstract:
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Markov Chain Monte Carlo (MCMC) is a powerful tool to sample from complex distributions. Due to the inherent randomness of the MCMC sampling, some privacy guarantees can be achieved. In the Bayesian framework, there exists work on the achieved upper bound for the privacy loss by posterior sampling in the setting of differential privacy (DP). In the implementation of the algorithms, the actual privacy loss can be adjusted indirectly through the subsampling ratio in the MCMC iterations or by changing the temperature of the target distribution. We propose a new differentially private Metropolis-Hastings algorithm by introducing auxiliary variables to help achieve DP explicitly. Compared to the approaches leveraging the inherent randomness to achieve DP, our approach can deliver privacy guarantees of different levels through tuning the parameters associated with the distribution of the auxiliary variables. Our empirical studies suggest the differentially private posterior distributions generated by our proposed method are closer to the true distributions at a given privacy budget in various settings compared to the existing methods for differentially private posterior sampling.
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