Experiments using machine learning to approximate likelihood ratios for mixture models

Likelihood ratio tests are a key tool in many fields of science. In order to evaluate the likelihood ratio the likelihood function is needed. However, it is common in fields such as High Energy Physics to have complex simulations that describe the distribution while not having a description of the likelihood that can be directly evaluated. In this setting it is impossible or computationally expensive to evaluate the likelihood. It is, however, possible to construct an equivalent version of the likelihood ratio that can be evaluated by using discriminative classifiers. We show how this can be used to approximate the likelihood ratio when the underlying distribution is a weighted sum of probability distributions (e.g. signal plus background model). We demonstrate how the results can be considerably improved by decomposing the ratio and use a set of classifiers in a pairwise manner on the components of the mixture model and how this can be used to estimate the unknown coefficients of the model, such as the signal contribution.