A Density-Ratio Readout Theory of Label-Free Out-of-Distribution Detection

Out-of-distribution (OOD) detection is not simply a test of nominal atypicality. Reference typicality need not reveal which distribution a sample was drawn from. An unlabeled test batch adds population-level evidence of how the test distribution differs from the reference. This paper asks what becomes detectable from this information and how much can be recovered from finite data. At the population level, the batch-to-reference density ratio identifies where the test population is overrepresented relative to the reference, providing direct evidence of distributional difference. Finite data, however, limit how finely this ratio can be resolved, and finer resolution requires higher-order batch moments. Low-degree readouts that maximize the deflection, a standardized batch–reference contrast, have closed forms in the batch moments under a whitened Gaussian working model, with degree 2 requiring only the batch mean and covariance. Experiments from CIFAR to ImageNet show that the simple degree-2 readout outperforms ratio estimators on average, with its near-OOD gain largely reproduced by the directions along which the batch covariance departs from the reference.