Statistic selection and MCMC for differentially private Bayesian estimation

This paper concerns differentially private Bayesian estimation of the parameters of a population distribution, when a statistic of a sample from that population is shared in noise to provide differential privacy. This work mainly addresses two problems: (1) What statistic of the sample should be shared privately? For the first question, i.e., the one about statistic selection, we promote using the Fisher information. We find out that, the statistic that is most informative in a non-privacy setting may not be the optimal choice under the privacy restrictions. We provide several examples to support that point. We consider several types of data sharing settings and propose several Monte Carlo-based numerical estimation methods for calculating the Fisher information for those settings. The second question concerns inference: (2) Based on the shared statistics, how could we perform effective Bayesian inference? We propose several Markov chain Monte Carlo (MCMC) algorithms for sampling from the posterior distribution of the parameter given the noisy statistic. The proposed MCMC algorithms can be preferred over one another depending on the problem. For example, when the shared statistics is additive and added Gaussian noise, a simple Metropolis-Hasting algorithm that utilizes the central limit theorem is a decent choice. We propose more advanced MCMC algorithms for several other cases of practical relevance. Our numerical examples involve comparing several candidate statistics to be shared privately. For each statistic, we perform Bayesian estimation based on the posterior distribution conditional on the privatized version of that statistic. We demonstrate that, the relative performance of a statistic, in terms of the mean squared error of the Bayesian estimator based on the corresponding privatized statistic, is adequately predicted by the Fisher information of the privatized statistic.