Objective Bayesian analysis of "on/off" measurements

Abstract: In high-energy astrophysics, it is common practice to account for the background overlaid with the counts from the source of interest with the help of auxiliary measurements carried on by pointing off-source. In this "on/off" measurement, one knows the number of photons detected while pointing to the source, the number of photons collected while pointing away of the source, and how to estimate the background counts in the source region from the flux observed in the auxiliary measurements. For very faint sources, the number of detected photons is so low that the approximations which hold asymptotically are not valid. On the other hand, the analytical solution exists for the Bayesian statistical inference, which is valid at low and high counts. Here we illustrate the objective Bayesian solution based on the reference posterior and compare the result with the approach very recently proposed by \citet{knoetig2014}, discussing its most delicate points. In addition, we propose to compute the significance of the excess with respect to the background-only expectation with a method which is able to account for any uncertainty on the background and is valid for any photon count. This method is compared to the widely used significance formula by \citet{LiMa83}, which is based on asymptotic properties.