  Searches for faint signals in counting experiments are often encountered in
particle physics and astrophysics, as well as in other fields. Many problems
can be reduced to the case of a model with independent and Poisson-distributed
signal and background. Often several background contributions are present at
the same time, possibly correlated. We provide the analytic solution of the
statistical inference problem of estimating the signal in the presence of
multiple backgrounds, in the framework of objective Bayes statistics. The model
can be written in the form of a product of a single Poisson distribution with a
multinomial distribution. The first is related to the total number of events,
whereas the latter describes the fraction of events coming from each individual
source. Correlations among different backgrounds can be included in the
inference problem by a suitable choice of the priors.
