  Identifying predictive factors for an outcome of interest via a multivariable
analysis is often difficult when the data set is small. Combining data from
different medical centers into a single (larger) database would alleviate this
problem, but is in practice challenging due to regulatory and logistic
problems. Federated Learning (FL) is a machine learning approach that aims to
construct from local inferences in separate data centers what would have been
inferred had the data sets been merged. It seeks to harvest the statistical
power of larger data sets without actually creating them. The FL strategy is
not always efficient and precise. Therefore, in this paper we refine and
implement an alternative Bayesian Federated Inference (BFI) framework for
multicenter data with the same aim as FL. The BFI framework is designed to cope
with small data sets by inferring locally not only the optimal parameter
values, but also additional features of the posterior parameter distribution,
capturing information beyond what is used in FL. BFI has the additional benefit
that a single inference cycle across the centers is sufficient, whereas FL
needs multiple cycles. We quantify the performance of the proposed methodology
on simulated and real life data.
