  The integration of data and knowledge from several sources is known as data
fusion. When data is only available in a distributed fashion or when different
sensors are used to infer a quantity of interest, data fusion becomes
essential. In Bayesian settings, a priori information of the unknown quantities
is available and, possibly, present among the different distributed estimators.
When the local estimates are fused, the prior knowledge used to construct
several local posteriors might be overused unless the fusion node accounts for
this and corrects it. In this paper, we analyze the effects of shared priors in
Bayesian data fusion contexts. Depending on different common fusion rules, our
analysis helps to understand the performance behavior as a function of the
number of collaborative agents and as a consequence of different types of
priors. The analysis is performed by using two divergences which are common in
Bayesian inference, and the generality of the results allows to analyze very
generic distributions. These theoretical results are corroborated through
experiments in a variety of estimation and classification problems, including
linear and nonlinear models, and federated learning schemes.
