Question: What is the meaning of 'honesty' in Algorithm 1?
Evidence: A growing amount of literature has been devoted to address the problem of HTE estimation with continuous treatments. The algorithms in the context of DRF include methods targeting at confounding bias, kernel-based or ML-based methods for regression bias reduction, and techniques of DRE or DML that balance the trade-off between the two biases. DRF combined with IPW can achieve consistent estimators by weighting the estimation with probability density of treatment. For the regression bias, employ parametric estimations and model DRF as non-parametric functions using kernel regression. To advance, propose a doubly robust estimators for DRF by combine the estimation of generalized propensity score with the estimation of outcome using kernel regression. Thus far, these approaches only provide global estimations and cannot handle massive amounts of high-dimensional data.
Recently, the great efficiency of ML methods motivates their generalization to the problem of HTE estimation. Towards that end, estimating DRF with ML-based algorithms is developed and among which tree-based models are a great candidate since it partitions the feature space for dimensionality reduction and maximizes the heterogeneity as well. Causal Forest (CF) proposed in utilizes a subset of training samples for growing trees by recursively partitioning via a splitting criterion. Then HTE estimation is given by weighted average over the outcomes of the remaining training samples, known as honesty principle. The final estimator obtained from CF further exhibits a lower degree of bias by tree ensembles. However, their splitting criterion relies on a linear model assumption and is formulated as the difference between the slopes of linear models. The same limitation applies to Orthogonal Random Forest and Bayesian Additive Regression Tress. Nevertheless, the complexity of HTE estimation may not be fully captured by linear or general parametric models, which brings the necessity of non-parametric ML models.
Kernel-based DML estimates the nuisance functions with cross-fitting, constructs a non-parametric DML estimator by Gaeutax Derivative. This motivates the utilization of DML and non-parametric estimation in our work, though it only provide global estimators with the limited capacity of localization. A fully non-parametric DML with locally weighted estimations for continuous treatments has not been considered yet.
Building on prior art, we point out that our integration of DRF estimation into CF overcome the challenges aforementioned by using local non-parametric DRF to constructing splitting criterion. We introduce the conceptual partial DRF as a component of the splitting criterion and employ the distance in the functional space of Partial DRF as a proxy for heterogeneity instead of the difference of slopes. Moreover, the partial DRF can be estimated precisely by the robust kernel-based DML estimators in our splitting criterion.
Then HTE estimation is given by weighted average over the outcomes of the remaining training samples, known as honesty principle.
Answer: We have included the details on causal trees and the associated honesty principle in the related work of the paper. To have an overview of the honesty principle, when we grow a causal tree, we only utilize a subset of training samples. The remaining training samples only contribute to computing the ATE estimators on each leaf node of the tree at the prediction stage. There is no intersection between the two subsets. This principle is vital in two aspects. On the one hand, it deals with overfitting and thereby achieving bias reduction. On the other side, it leads to the convergence of the estimator.