  Cosmic rays (CRs) are protons and atomic nuclei that flow into our Solar
system and reach the Earth with energies of up to ~10^21 eV. The sources of
ultra-high energy cosmic rays (UHECRs) with E >~ 10^19 eV remain unknown,
although there are theoretical reasons to think that at least some come from
active galactic nuclei (AGNs). One way to assess the different hypotheses is by
analysing the arrival directions of UHECRs, in particular their
self-clustering. We have developed a fully Bayesian approach to analyzing the
self-clustering of points on the sphere, which we apply to the UHECR arrival
directions. The analysis is based on a multi-step approach that enables the
application of Bayesian model comparison to cases with weak prior information.
We have applied this approach to the 69 highest energy events recorded by the
Pierre Auger Observatory (PAO), which is the largest current UHECR data set. We
do not detect self-clustering, but simulations show that this is consistent
with the AGN-sourced model for a data set of this size. Data sets of several
hundred UHECRs would be sufficient to detect clustering in the AGN model.
Samples of this magnitude are expected to be produced by future experiments,
such as the Japanese Experiment Module Extreme Universe Space Observatory
(JEM-EUSO).
