  There is significant growth and interest in the use of synthetic data as an
enabler for machine learning in environments where the release of real data is
restricted due to privacy or availability constraints. Despite a large number
of methods for synthetic data generation, there are comparatively few results
on the statistical properties of models learnt on synthetic data, and fewer
still for situations where a researcher wishes to augment real data with
another party's synthesised data. We use a Bayesian paradigm to characterise
the updating of model parameters when learning in these settings, demonstrating
that caution should be taken when applying conventional learning algorithms
without appropriate consideration of the synthetic data generating process and
learning task. Recent results from general Bayesian updating support a novel
and robust approach to Bayesian synthetic-learning founded on decision theory
that outperforms standard approaches across repeated experiments on supervised
learning and inference problems.
