Parameter uncertainties in weighted unbinned maximum likelihood fits

TL;DR

This paper addresses the challenge of accurately estimating parameter uncertainties in weighted unbinned maximum likelihood fits, especially in high energy physics, by deriving correct asymptotic formulas and analyzing common methods.

Contribution

It introduces asymptotically correct expressions for parameter uncertainties in weighted likelihood fits and evaluates their performance against existing approaches.

Findings

Naive second-derivative method often incorrect for weighted fits.

Derived formulas improve confidence interval coverage.

Uncertainties in event weights and nuisance parameters are analyzed.

Abstract

Parameter estimation via unbinned maximum likelihood fits is central for many analyses performed in high energy physics. Unbinned maximum likelihood fits using event weights, for example to statistically subtract background contributions via the sPlot formalism, or to correct for acceptance effects, have recently seen increasing use in the community. However, it is well known that the naive approach to the estimation of parameter uncertainties via the second derivative of the logarithmic likelihood does not yield confidence intervals with the correct coverage in the presence of event weights. This paper derives the asymptotically correct expressions and compares them with several commonly used approaches for the determination of parameter uncertainties, some of which are shown to not generally be asymptotically correct. In addition, the effect of uncertainties on event weights is discussed, including uncertainties that can arise from the presence of nuisance parameters in the determination of sWeights.