Calibrated Inference for the Conditional Average Treatment Effect in the Few-Placebo Regime via Gaussian Processes

In the few-placebo regime ($N_0\ll N_1$), the control-arm outcome surface ${\mu }_0$ is only weakly identified, and this under-identification is a primary determinant of interval reliability for the conditional average treatment effect (CATE). Existing estimators can under-cover because their inference procedures do not propagate this under-identification into the interval: point-estimate methods inherit nuisance bias, and GP methods with MAP hyperparameter estimation suppress residual uncertainty in the treatment-effect hyperparameters. We propose ISO-HMC, a structured Gaussian-process model that places a separate prior on the treatment-effect function and marginalises over its hyperparameters $({\sigma }_{\tau },{\ell }_{\tau },{\sigma }^2)$ via HMC. By propagating this uncertainty through the law of total variance, ISO-HMC achieves calibration when the treatment effect is smooth and low-dimensional. We test three claims: (i) ISO-HMC is calibrated in its target regime; (ii) it achieves this without sacrificing interval informativeness; (iii) it degrades gracefully under misspecification, with an independent-arm GP as a safe fallback. All three are supported: on smooth low-dimensional designs ISO-HMC attains calibrated coverage and substantially narrower intervals than a conservative independent-arm GP baseline, while on nonlinear or high-dimensional designs it under-covers and the independent-arm GP restores calibration. Overall, these results show that in the few-placebo regime, calibrated CATE intervals benefit from propagating uncertainty through both the outcome model and the treatment-effect hyperparameters. Suppressing either through point estimates or MAP leads to overconfidence and under-coverage.