Every generative model eventually fails a large enough test: SIGMA profiles fit uncertainty and keeps the same model trustworthy on test sets an order of magnitude larger

When a generative model serves as the reference density of a physics analysis, validation needs more than a goodness-of-fit test: the fit's own uncertainty has to enter the statistical test. Earlier validation treated the fitted model as an exact point null, so a model fitted to a finite training set necessarily differs from the truth and the test rejects it once the test set can resolve that difference. In a one-dimensional standard-normal example with a perfectly specified model family, the fitted model is indistinguishable from the truth at small test sizes but is rejected with high significance once the test sample exceeds the training sample, and any consistent goodness-of-fit test shows the same trend.

SIGMA describes the density with a Gaussian mixture model and characterizes parameter uncertainty in the tangent space spanned by its score functions around the best fit. Neural density estimators have many non-identifiable parameters and no tractable parametrized uncertainty, while ensemble uncertainties span only the weights of the trained members. Gaussian mixture parameters are identifiable up to relabeling, and their scores and Fisher information are analytic, giving the asymptotic maximum-likelihood covariance and turning the test into a convex problem.

The test profiles the fit uncertainty as a Gaussian-constrained nuisance parameter, turning the null from a point hypothesis into a composite one so that a rejection reflects only what the model's uncertainty cannot explain. The conventional Neyman–Pearson likelihood-ratio test treats the reference model as exact, so its outcome is driven by the relation between test and training sample sizes rather than by model quality. In the two-dimensional toy model the point-null median p-value starts degrading once the test set exceeds the training size and collapses to zero for every number of mixture components, while the composite null with parametric-bootstrap calibration stays calibrated for test sets far larger than the training size.

On a four-dimensional contrastive embedding of QCD multijets from JetClass, SIGMA models the background density and yields closure results. Latent-space anomaly detection needs a reliable reference density, and large simulated samples are expensive, so a background estimate with statistical guarantees was missing. The point-null test starts degrading before the test set reaches the training size, the Gaussian prescription holds calibration to a larger test set, and the bootstrap prescription holds it further still, an order of magnitude beyond the point null.

The demonstrated scope is a two-dimensional toy model with known analytic truth and background modeling of QCD multijets in a four-dimensional contrastive embedding of the JetClass dataset; the authors state the method suits low-dimensional spaces and suggest compressing high-dimensional data into a few summary statistics first.

The method covers only the statistical uncertainty from finite training samples and optimization stochasticity, not model bias, so a model family that cannot represent the data is still rejected by a sufficiently powerful test; the Gaussian prescription requires the asymptotic regime, the bootstrap prescription relies on the fit spread being similar for the truth and nearby densities, and bootstrap calibration breaks down when the kernel width is too narrow; the number of components and the truncation hyperparameter are chosen empirically, and closure tests need held-out samples as large as the intended inference dataset.