# Goodness-of-Fit Testing

A generative model fitted to a finite training set is never exactly the true density, so any sufficiently powerful goodness-of-fit test will eventually reject it once the test sample is large enough, even when the model family is correct. SIGMA writes the parameter uncertainty of a Gaussian mixture model as a tractable covariance from its score functions and Fisher information, then promotes the conventional point-null test to a composite null in which that uncertainty is profiled as a nuisance parameter. On a two-dimensional toy model and on a QCD background benchmark in a learned jet embedding, including the uncertainty keeps the same model trustworthy on test sets an order of magnitude larger than the standard test allows.

A generative model fitted to a finite training set is never exactly the true density, so any sufficiently powerful goodness-of-fit test will eventually reject it once the test sample is large enough, even when the model family is correct. SIGMA writes the parameter uncertainty of a Gaussian mixture model as a tractable covariance from its score functions and Fisher information, then promotes the conventional point-null test to a composite null in which that uncertainty is profiled as a nuisance parameter. On a two-dimensional toy model and on a QCD background benchmark in a learned jet embedding, including the uncertainty keeps the same model trustworthy on test sets an order of magnitude larger than the standard test allows.