SIMBAD references

We present a Bayesian hierarchical model which enables a joint fit of the ultrahigh-energy cosmic ray (UHECR) energy spectrum and arrival directions within the context of a physical model for the UHECR phenomenology. In this way, possible associations with astrophysical source populations can be assessed in a physically and statistically principled manner. The importance of including the UHECR energy data and detection effects is demonstrated through simulation studies, showing that the effective GZK horizon is significantly extended for typical reconstruction uncertainties. We also verify the ability of the model to fit and recover physical parameters from CRPROPA 3 simulations. Finally, the model is used to assess the fraction of the publicly available data set of 231 UHECRs detected by the Pierre Auger Observatory which are associated with the Fermi-LAT 2FHL catalogue, a set of starburst galaxies, and Swift-BAT hard X-ray sources. We find association fractions of 9.5^+2.4_–5.9, 22.7^+6.6_–12.4, and 22.8^+6.6_–8.0 per cent for the 2FHL, starburst galaxies, and Swift-BAT catalogues respectively.