  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.
