  In this work, we introduce a differentially private method for generating
synthetic data from vertically partitioned data, \emph{i.e.}, where data of the
same individuals is distributed across multiple data holders or parties. We
present a differentially privacy stochastic gradient descent (DP-SGD) algorithm
to train a mixture model over such partitioned data using variational
inference. We modify a secure multiparty computation (MPC) framework to combine
MPC with differential privacy (DP), in order to use differentially private MPC
effectively to learn a probabilistic generative model under DP on such
vertically partitioned data.
  Assuming the mixture components contain no dependencies across different
parties, the objective function can be factorized into a sum of products of the
contributions calculated by the parties. Finally, MPC is used to compute the
aggregate between the different contributions. Moreover, we rigorously define
the privacy guarantees with respect to the different players in the system. To
demonstrate the accuracy of our method, we run our algorithm on the Adult
dataset from the UCI machine learning repository, where we obtain comparable
results to the non-partitioned case.
