2015ApJ...798....5C

In high-energy astrophysics, it is common practice to account for the background overlaid with counts from the source of interest with the help of auxiliary measurements carried out by pointing off-source. In this "on/off" measurement, one knows the number of photons detected while pointing toward the source, the number of photons collected while pointing away from 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 photons detected is so low that the approximations that hold asymptotically are not valid. On the other hand, an 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 Knoetig, and discuss 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 that 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 Li & Ma, which is based on asymptotic properties.