Targeted Maximum Likelihood Estimation for Prediction Calibration

Estimators of the conditional expectation, i.e., prediction, function involve a global bias-variance trade off. In some cases, an estimator that yields unbiased estimates of the conditional expectation for a particular partitioning of the data may be desirable. Such estimators are calibrated with respect to the partitioning. We identify the conditional expectation given a particular partitioning as a smooth parameter of the distribution of the data, where the partitioning may be defined on the covariate space or on the prediction space of the estimator. We propose a targeted maximum likelihood estimation (TMLE) procedure that updates an initial prediction estimator such that the updated estimator yields an unbiased and efficient estimator of this smooth parameter in the nonparametric statistical model. When the partitioning is defined on the prediction space of the estimator, our TMLE involves enforcing an implicit constraint on the estimator itself. We show that our resulting estimator of the smooth parameter is equal to the empirical estimator, which is also known to be unbiased and efficient in the nonparametric statistical model. We derive the TMLE for single time-point prediction and also time-dependent prediction in a counting process framework.