  The Earth is continuously showered by charged cosmic ray particles, naturally
produced atomic nuclei moving with velocity close to the speed of light. Among
these are ultra high energy cosmic ray particles with energy exceeding 5x10^19
eV, which is ten million times more energetic than the most energetic particles
produced at the Large Hadron Collider. Astrophysical questions include: what
phenomenon accelerates particles to such high energies, and what sort of nuclei
are energized? Also, the magnetic deflection of the trajectories of the cosmic
rays makes them potential probes of galactic and intergalactic magnetic fields.
We develop a Bayesian hierarchical model that can be used to compare different
association models between the cosmic rays and source population, using Bayes
factors. A measurement model with directional uncertainties and accounting for
non-uniform sky exposure is incoporated into the model. The methodology allows
us to learn about astrophysical parameters, such as those governing the source
luminosity function and the cosmic magnetic field.
