  Estimating the probability density of a population while preserving the
privacy of individuals in that population is an important and challenging
problem that has received considerable attention in recent years. While the
previous literature focused on frequentist approaches, in this paper, we
propose a Bayesian nonparametric mixture model under differential privacy (DP)
and present two Markov chain Monte Carlo (MCMC) algorithms for posterior
inference. One is a marginal approach, resembling Neal's algorithm 5 with a
pseudo-marginal Metropolis-Hastings move, and the other is a conditional
approach. Although our focus is primarily on local DP, we show that our MCMC
algorithms can be easily extended to deal with global differential privacy
mechanisms. Moreover, for some carefully chosen mechanisms and mixture kernels,
we show how auxiliary parameters can be analytically marginalized, allowing
standard MCMC algorithms (i.e., non-privatized, such as Neal's Algorithm 2) to
be efficiently employed. Our approach is general and applicable to any mixture
model and privacy mechanism. In several simulations and a real case study, we
discuss the performance of our algorithms and evaluate different privacy
mechanisms proposed in the frequentist literature.
