Adaptive Targeted Maximum Likelihood Estimation of the Mean Potential Outcome under a Treatment Rule

Estimating the mean counterfactual outcome under a treatment rule is a central problem in causal inference and policy evaluation. Standard estimators, including inverse probability weighting (IPW), augmented IPW (AIPW), and targeted maximum likelihood estimation (TMLE), can become unstable under practical positivity violations because their targeting or weighting steps depend on inverse propensity scores. We propose an adaptive targeted maximum likelihood estimation (A-TMLE) framework that uses a data-adaptive working model for the conditional average treatment effect (CATE). This working model induces a projected policy-value parameter, which coincides with the nonparametric mean potential outcome when the CATE is well represented by the adaptive basis. We derive the efficient influence function for the projected parameter and characterize its second-order remainder. We also introduce a regularized TMLE that targets the nonparametric policy value using a stabilized targeting covariate obtained by projecting the standard TMLE clever covariate onto the score space induced by the CATE working model. We quantify the first-order plug-in bias of regularized TMLE relative to the nonparametric target. The resulting targeting steps avoid direct inverse propensity score weighting, improving stability under limited overlap. In simulations, A-TMLE and regularized TMLE achieve lower mean squared error and improved coverage compared with IPW, AIPW, and standard TMLE under practical positivity violations, while remaining competitive when treatment overlap is strong. A real-data application to the Right Heart Catheterization study illustrates that the adaptive estimators produce stable policy-value estimates with substantially shorter confidence intervals than IPW and AIPW.