We study identification of a structural group effect when the group indicator $G\in\{0,1\}$ is unobserved but the analyst observes a calibrated probability score $p$ satisfying $E[G\mid p,X]=p$. Under a constant-coefficient structural mean model, the latent-group coefficient $τ$ is point-identified by a closed-form ratio of observable moments whose denominator is the residual score variance $V^{*}=E[(p-E[p\mid X])^2]$. Identification fails exactly when the score is a deterministic function of $X$; we construct an explicit continuum of observationally equivalent models showing the failure is genuine. The marginal latent mean gap decomposes as $τ$ plus a compositional term that is itself identified in closed form, and we characterise when the two coincide. The oracle estimator is $\sqrt{n}$-consistent and asymptotically normal with a closed-form sandwich variance. Under calibration error bounded by $δ$, the bias obeys a sharp bound proportional to $δ/V^{*}$, and hard-threshold classification attenuates the estimated gap. Monte Carlo experiments confirm the theory, including the variance-weighted estimand under heterogeneous effects.
A Kernel-Based Conditional Two-Sample Test Using Nearest Neighbors (with Applications to Calibration, Regression Curves, and Simulation-Based Inference)
Correcting for confounding in longitudinal experiments: positioning non-linear mixed effects modeling as implementation of standardization using latent conditional exchangeability
Towards Instance-Wise Calibration: Local Amortized Diagnostics and Reshaping of Conditional Densities (LADaR)
1 Department of Physics；Astronomy, University of Pittsburgh；Cosmology Center (PITT PACC), University of Pittsburgh；3 Department of Statistical Sciences, University of Toronto；4 Canadian Institute for Theoretical Astrophysics (CITA), University of Toronto；5 Dunlap Institute for Astronomy \& Astrophysics, University of Toronto；6 Vector Institute；7 Department of Statistics；Data Science, Carnegie Mellon University；8 Department of Statistics, Federal University of S\ a o Carlos (UFSCar)
