Constraining duty cycles through a Bayesian technique
The duty cycle (DC) of astrophysical sources is generally defined as the fraction of time during which the sources are active. However, DCs are generally not provided with statistical uncertainties, since the standard approach is to perform Monte Carlo bootstrap simulations to evaluate them, which can be quite time consuming for a large sample of sources. As an alternative, considerably less time-consuming approach, we derived the theoretical expectation value for the DC and its error for sources whose state is one of two possible, mutually exclusive states, inactive (off) or flaring (on), as based on a finite set of independent observational data points. Following a Bayesian approach, we derived the analytical expression for the posterior, the conjugated distribution adopted as prior, and the expectation value and variance. We applied our method to the specific case of the inactivity duty cycle (IDC) for supergiant fast X-ray transients. We also studied IDC as a function of the number of observations in the sample. Finally, we compare the results with the theoretical expectations. We found excellent agreement with our findings based on the standard bootstrap method. Our Bayesian treatment can be applied to all sets of independent observations of two-state sources, such as active galactic nuclei, X-ray binaries, etc. In addition to being far less time consuming than bootstrap methods, the additional strength of this approach becomes obvious when considering a well-populated class of sources ($N_{\rm src} \geq 50$) for which the prior can be fully characterized by fitting the distribution of the observed DCs for all sources in the class, so that, through the prior, one can further constrain the DC of a new source by exploiting the information acquired on the DC distribution derived from the other sources. [Abridged]