A General Theory of Goodness of Fit in Likelihood Fits
msra(2005)
摘要
Maximum likelihood fits to data can be performed using binned data and
unbinned data. The likelihood fits in either case produce only the fitted
quantities but not the goodness of fit. With binned data, one can obtain a
measure of the goodness of fit by using the $\chi^2$ method, after the maximum
likelihood fitting is performed. With unbinned data, currently, the fitted
parameters are obtained but no measure of goodness of fit is available. This
remains, to date, an unsolved problem in statistics. By considering the
transformation properties of likelihood functions with respect to change of
variable, we conclude that the likelihood ratio of the theoretically predicted
probability density to that of {\it the data density} is invariant under change
of variable and provides the goodness of fit. We show how to apply this
likelihood ratio for binned as well as unbinned likelihoods and show that even
the $\chi^2$ test is a special case of this general theory. In order to
calculate errors in the fitted quantities, we need to solve the problem of
inverse probabilities. We use Bayes' theorem to do this, using the data density
obtained in the goodness of fit. This permits one to invert the probabilities
without the use of a Bayesian prior. The resulting statistics is consistent
with frequentist ideas.
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关键词
maximum likelihood,goodness of fit,data analysis,likelihood ratio,high energy physics,likelihood function,probability density
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