2014ApJ...790..106K

For decades researchers have studied the On/Off counting problem where a measured rate consists of two parts. One part is due to a signal process and the other is due to a background process, the magnitudes for both of which are unknown. While most frequentist methods are adequate for large number counts, they cannot be applied to sparse data. Here, I want to present a new objective Bayesian solution that only depends on three parameters: the number of events in the signal region, the number of events in the background region, and the ratio of the exposure for both regions. First, the probability of the counts only being due to background is derived analytically. Second, the marginalized posterior for the signal parameter is also derived analytically. With this two-step approach it is easy to calculate the signal's significance, strength, uncertainty, or upper limit in a unified way. This approach is valid without restrictions for any number count, including zero, and may be widely applied in particle physics, cosmic-ray physics, and high-energy astrophysics. In order to demonstrate the performance of this approach, I apply the method to gamma-ray burst data.