Coarsening Latent-Class Probabilities: Directional Distortion and Coverage Loss

Outcomes are increasingly regressed on a calibrated probability vector for unobserved class membership, and that vector is often coarsened to a hard label first. Under a constant-coefficient structural mean and conditional calibration, the observed-data problem is a partially linear regression of the outcome on the probability vector; we take this reduction as the starting point and ask what coarsening costs. For any coarsening, the plug-in estimator converges to $\mathcal{A}τ$, where the coarsening operator satisfies $\mathcal{A}=I+D^{-1}\mathbb{E}[a_{h}u^{\top}]$ with $u$ the discarded signal. Coarsening is therefore free exactly when what is discarded is uncorrelated with what is kept, and is otherwise anisotropic: it distorts some contrasts far more than others. The same operator governs inference. The Wald interval built from coarsened labels has limiting coverage $Φ(z-λ)-Φ(-z-λ)$, with $λ$ the ratio of the coarsening bias to the reported standard error; because $\mathcal{A}$ and that standard error depend on observables alone, the coverage implied by the estimated index can be approximated before the interval is reported. Simulations show severe coverage loss after argmax coarsening, and three real-data audits exhibit the direction-specific distortion that hard labels induce.