Chi-square Fitting When Overall Normalization is a Fit Parameter

TL;DR

This paper explains why fitting event distributions with an unknown total normalization using chi-square can lead to biased results, and advocates for maximum likelihood as a better solution, illustrating this with simple examples.

Contribution

It clarifies the cause of normalization bias in chi-square fits and demonstrates how maximum likelihood provides an elegant, simple solution, including a modification to chi-square that mimics likelihood.

Findings

Maximum likelihood corrects normalization bias in fits.

A simple chi-square modification replicates maximum likelihood results.

The paper provides didactic explanation and illustrative examples.

Abstract

The problem of fitting an event distribution when the total expected number of events is not fixed, keeps appearing in experimental studies. In a chi-square fit, if overall normalization is one of the parameters parameters to be fit, the fitted curve may be seriously low with respect to the data points, sometimes below all of them. This problem and the solution for it are well known within the statistics community, but, apparently, not well known among some of the physics community. The purpose of this note is didactic, to explain the cause of the problem and the easy and elegant solution. The solution is to use maximum likelihood instead of chi-square. The essential difference between the two approaches is that maximum likelihood uses the normalization of each term in the chi-square assuming it is a normal distribution, 1/sqrt(2 pi sigma-square). In addition, the normalization is applied to the theoretical expectation not to the data. In the present note we illustrate what goes wrong and how maximum likelihood fixes the problem in a very simple toy example which illustrates the problem clearly and is the appropriate physics model for event histograms. We then note how a simple modification to the chi-square method gives a result identical to the maximum likelihood method.