A fast and stable approximation to the Barlow-Beeston template fit

In their paper from 1993, Barlow and Beeston presented an exact likelihood for the problem of fitting a composite model consisting of binned templates obtained from Monte-Carlo simulation which are fitted to equally binned data. Solving the exact likelihood is a technical challenge, and so Conway in 2011 proposed an approximated likelihood that overcomes them. Conway's approximate likelihood and a new approximate likelihood are derived from the exact one in this paper. The new approximation is expected perform better than Conway's. The likelihoods are extended to the problem of fitting weighted data and weighted templates. The performance of estimates obtained with all three likelihoods are studied on a toy example. The exact likelihood performs best, closely followed by the new approximate likelihood, putting Conway's likelihood in third place.