# Identification of Latent Group Effects under Conditional Calibration

arXiv: https://arxiv.org/abs/2604.08798 · HTML: https://mlsanity.com/paper/2604.08798/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2026.

## Built on by (most cited first, 2 of 2)

- [When to Trust Confidence Thresholding: Calibration Diagnostics for Pseudo-Labelled Regression](https://mlsanity.com/paper/2605.12780/) (2026)
- [Coarsening Latent-Class Probabilities: Directional Distortion and Coverage Loss](https://mlsanity.com/paper/2608.11784/) (2026)

## Abstract (arXiv)

We study identification of a structural group effect when the group indicator G∈0,1 is unobserved, but the analyst observes a calibrated probability score p satisfying E[Gmid 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[pmid 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 √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.