  Collaborative data analysis between countries is crucial for enabling fast
responses to increasingly multi-country disease outbreaks. Often, data early in
outbreaks are of sensitive nature and subject to strict privacy restrictions.
Thus, federated analysis, which implies decentralised collaborative analysis
where no raw data sharing is required, emerged as a novel approach solving
issues around data privacy and confidentiality. In the present study, we
propose two approaches to federated analysis, based on simple Bayesian
statistics and exploit this simplicity to make them feasible for rapid
collaboration without the risks of data leaks and data reidentification, as
they require neither data sharing nor direct communication between devices. The
first approach uses summaries from parameters' posteriors previously obtained
at a different location to update truncated normal distributions approximating
priors of a new model. The second approach uses the entire previously sampled
posterior, approximating via a multivariate normal distribution. We test these
models on simulated and on real outbreak data to estimate the incubation period
of infectious diseases. Results indicate that both approaches can recover
incubation period parameters accurately, but they differ in terms of
inferential capacity. The posterior summary approach shows higher stability and
precision, but it cannot capture posterior correlations, meaning it is
inferentially limited. The whole posterior approach can capture correlations,
but it shows less stability, and its applicability is limited to fewer prior
distributions. We discuss results in terms of the advantages of their
simplicity and privacy-preserving properties, and in terms of their limited
generalisability to more complex analytical models.
