Incorporating Nuisance Parameters in Likelihoods for Multisource Spectra

TL;DR

The paper addresses fitting multisource spectra under systematic uncertainties by embedding nuisance parameters into a Poisson likelihood and employing the profile likelihood for efficient inference. It introduces three main nuisance types—multiplicative factors, shape morphing parameters, and statistical uncertainties in predicted spectra—with explicit formulations and practical remedies. Key methods include Gaussian constraints for multiplicative uncertainties, quadratic-interpolation morphing for shape distortions, and a Bar lon-Beeston–style treatment for per-bin statistical uncertainties, alongside practical safeguards to maintain numerical stability. The approach offers a unified, computationally efficient framework for exclusions, discoveries, and precise measurements in particle-physics spectral analyses.

Abstract

We describe here the general mathematical approach to constructing likelihoods for fitting observed spectra in one or more dimensions with multiple sources, including the effects of systematic uncertainties represented as nuisance parameters, when the likelihood is to be maximized with respect to these parameters. We consider three types of nuisance parameters: simple multiplicative factors, source spectra "morphing" parameters, and parameters representing statistical uncertainties in the predicted source spectra.