  Differential privacy (DP) has been accepted as a rigorous criterion for
measuring the privacy protection offered by random mechanisms used to obtain
statistics or, as we will study here, synthetic datasets from confidential
data. Methods to generate such datasets are increasingly numerous, using varied
tools including Bayesian models, deep neural networks and copulas. However,
little is still known about how to properly perform statistical inference with
these differentially private synthetic (DIPS) datasets. The challenge is for
the analyses to take into account the variability from the synthetic data
generation in addition to the usual sampling variability. A similar challenge
also occurs when missing data is imputed before analysis, and statisticians
have developed appropriate inference procedures for this case, which we tend
extended to the case of synthetic datasets for privacy. In this work, we study
the applicability of these procedures, based on combining rules, to the
analysis of DIPS datasets. Our empirical experiments show that the proposed
combining rules may offer accurate inference in certain contexts, but not in
all cases.
