Hard-label shortcuts in AI regression cause severe coverage loss, study shows

Marcell T. Kurbucz's working paper, "Coarsening Latent-Class Probabilities: Directional Distortion and Coverage Loss," posted on arXiv (2608.11784), examines a widespread practice in applied machine learning and econometrics: taking a calibrated probability vector for unobserved class membership and reducing it to a single hard label before regressing an outcome on that label. Under a constant-coefficient structural mean and conditional calibration, the author derives a partially linear regression reduction and then quantifies exactly what this coarsening step costs. The central result is that for any coarsening rule, the plug-in estimator converges to Aτ, where the coarsening operator A equals I + D⁻¹E[a_h uᵀ], with u being the discarded signal. Coarsening is therefore only free when what is thrown away is uncorrelated with what is kept. Otherwise, the distortion is anisotropic—some contrasts suffer far more degradation than others—and no simple isotropic adjustment fixes it.

The same operator also undermines inference. The Wald interval built from coarsened labels has limiting coverage of Φ(z−λ) − Φ(−z−λ), where λ is the ratio of the coarsening bias to the reported standard error. Because both A and that standard error depend only on observables, the approximate coverage can be computed in advance from the estimated index—before the interval is even reported. Simulations demonstrate severe coverage loss after argmax coarsening, and three real-world audits (including a land-cover classification application) exhibit the direction-specific distortion that hard labels induce. The paper, running 43 pages with 6 figures and 11 tables, includes supplementary proofs, two additional identification results, and further experiments. For practitioners, the lesson is clear: coarsening probability vectors to hard labels is not a harmless preprocessing step—it can silently bias causal estimates and invalidate confidence intervals. The author provides tools to diagnose and anticipate this degradation using only observed data.