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 $p\in[0,1]$ satisfying $\mathbb{E}[G|p,X]=p$. Under a constant-coefficient structural mean model, the latent-group coefficient $τ$ is point-identified from the joint law of observables $(Y,X,p)$ by a simple ratio of weighted moments: the covariance of the signed score $2p-1$ with the covariate-partialled outcome, divided by twice the residual variance of the score after conditioning on covariates. Identification fails if and only if the score is a deterministic function of $X$; we establish this by constructing an explicit continuum of observationally equivalent models indexed by arbitrary values of $τ$. The identified coefficient differs from the marginal latent mean gap by a compositional term that is unidentified without further assumptions; we give a necessary and sufficient condition for the two to coincide. The oracle estimator is $\sqrt{n}$-consistent and asymptotically normal with a closed-form sandwich variance. Under calibration error bounded uniformly by $δ$, the bias is bounded by $|τ|\,\mathbb{E}[|2p-1|]\,δ\,(2V^*)^{-1}$, a bound that is sharp over all calibration error functions of that magnitude. Hard-threshold classification at $p=1/2$ attenuates the estimated gap by a factor strictly less than one. Monte Carlo experiments confirm the asymptotic theory, trace the divergence of RMSE as $V^*\to 0$, illustrate the attenuation bias of hard-threshold classification, and verify identification of the variance-weighted estimand under heterogeneous effects.

翻译：我们研究当分组指标 $G\in\{0,1\}$ 未被观测，但分析者观察到满足 $\mathbb{E}[G|p,X]=p$ 的校准概率得分 $p\in[0,1]$ 时，结构性组效应的识别问题。在常系数结构性均值模型下，潜在组系数 $τ$ 可通过可观测变量 $(Y,X,p)$ 的联合分布由一个简单的加权矩比率点识别：即符号得分 $2p-1$ 与协变量部分化结果之间的协方差，除以得分在条件于协变量后的残差方差的两倍。当且仅当得分是 $X$ 的确定性函数时，识别失败；我们通过构造一个由任意 $τ$ 值索引的显式连续观测等价模型来证明这一点。识别出的系数与边际潜在均值差距相差一个在无进一步假设下无法识别的组成项；我们给出了两者相等的必要充分条件。神谕估计量是 $\sqrt{n}$ 一致且渐近正态的，具有封闭形式的夹心方差。在校准误差均匀有界于 $δ$ 的条件下，偏差由 $|τ|\,\mathbb{E}[|2p-1|]\,δ\,(2V^*)^{-1}$ 界定，该界限在该量级的所有校准误差函数上是尖锐的。在 $p=1/2$ 处的硬阈值分类使估计差距衰减一个严格小于1的因子。蒙特卡洛实验证实了渐近理论，追踪了当 $V^*\to 0$ 时RMSE的发散性，说明了硬阈值分类的衰减偏差，并验证了异质性效应下方差加权估计量的可识别性。