A new maximum-likelihood method for template fits

Abstract

A common statistical problem in particle physics is to extract the number of samples which originate from a statistical process in an ensemble containing a mix of several contributing processes. The probability density function of each process is usually not exactly known. Barlow and Beeston found an exact likelihood for the problem of fitting binned templates obtained from Monte-Carlo simulation to binned data, which propagates the uncertainty of the templates into the result. Solving the exact likelihood is technically challenging, however. The original paper also did not provide a way to use weighted simulation samples with varying weights. Other papers have introduced alternative likelihoods to address these points. In this paper, a new approximate likelihood is derived from the exact Barlow–Beeston one. The new likelihood is generalized to fits of weighted templates to weighted data. The performance of the new likelihood is evaluated based on toy examples. The performance is excellent – point estimates have small bias and confidence intervals have good coverage – and is comparable to the exact Barlow–Beeston likelihood when the templates are not weighted. The new likelihood evaluates faster than the Barlow–Beeston one when the number of bins is large.