  Generating synthetic data, with or without differential privacy, has
attracted significant attention as a potential solution to the dilemma between
making data easily available, and the privacy of data subjects. Several works
have shown that consistency of downstream analyses from synthetic data,
including accurate uncertainty estimation, requires accounting for the
synthetic data generation. There are very few methods of doing so, most of them
for frequentist analysis. In this paper, we study how to perform consistent
Bayesian inference from synthetic data. We prove that mixing posterior samples
obtained separately from multiple large synthetic data sets converges to the
posterior of the downstream analysis under standard regularity conditions when
the analyst's model is compatible with the data provider's model. We also
present several examples showing how the theory works in practice, and showing
how Bayesian inference can fail when the compatibility assumption is not met,
or the synthetic data set is not significantly larger than the original.
