  Marginal-based methods achieve promising performance in the synthetic data
competition hosted by the National Institute of Standards and Technology
(NIST). To deal with high-dimensional data, the distribution of synthetic data
is represented by a probabilistic graphical model (e.g., a Bayesian network),
while the raw data distribution is approximated by a collection of
low-dimensional marginals. Differential privacy (DP) is guaranteed by
introducing random noise to each low-dimensional marginal distribution. Despite
its promising performance in practice, the statistical properties of
marginal-based methods are rarely studied in the literature. In this paper, we
study DP data synthesis algorithms based on Bayesian networks (BN) from a
statistical perspective. We establish a rigorous accuracy guarantee for
BN-based algorithms, where the errors are measured by the total variation (TV)
distance or the $L^2$ distance. Related to downstream machine learning tasks,
an upper bound for the utility error of the DP synthetic data is also derived.
To complete the picture, we establish a lower bound for TV accuracy that holds
for every $\epsilon$-DP synthetic data generator.
