Appendix D - Fits including systematic errors

Summary

The maximum likelihood method and least squares fits

The likelihood function L is defined as the a priori probability of a given outcome. Since Gaussian probability distributions are common (as a consequence of the central limit theorem) and since there is also an important theorem regarding likelihood ratios for composite hypotheses, it is convenient to use the logarithm of the likelihood, W = -2 ln L.

There are generally two types of fits used for data from experiments. (i) The standard maximum likelihood method is to fit a function to n independent trials of the same distribution, for example the muon lifetime measured from the decay times t_i of individual decays. (ii) The method of least squares is also an extremum method which performs Gaussian least squares fits for binned data from histograms to predictions of the expectation values at each data point, which may be functions of parameters to be determined. This will be discussed following a review of the standard maximum likelihood method and the likelihood ratio test.

Maximum likelihood fit to a Gaussian, an instructive example

A nice example of the standard maximum likelihood method is a fit to a Gaussian with mean μ and variance σ² from the observation of n independent trials with values x_i from this distribution, in other words a sample of n trials from this population.