  We study how to communicate findings of Bayesian inference to third parties,
while preserving the strong guarantee of differential privacy. Our main
contributions are four different algorithms for private Bayesian inference on
proba-bilistic graphical models. These include two mechanisms for adding noise
to the Bayesian updates, either directly to the posterior parameters, or to
their Fourier transform so as to preserve update consistency. We also utilise a
recently introduced posterior sampling mechanism, for which we prove bounds for
the specific but general case of discrete Bayesian networks; and we introduce a
maximum-a-posteriori private mechanism. Our analysis includes utility and
privacy bounds, with a novel focus on the influence of graph structure on
privacy. Worked examples and experiments with Bayesian na{\"i}ve Bayes and
Bayesian linear regression illustrate the application of our mechanisms.
