Title: Bayesian Approaches to Collaborative Data Analysis with Strict Privacy Restrictions

Abstract: 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.