I am reading Athey paper on generalized random forests that you can find for instance here https://arxiv.org/abs/1610.01271.
Now I do not understand formula (6) where it is defined an approximation of the estimate of the treatment effect theta on the child node C:
As far as I have understood the approximation is defined as the estimate of the treatment effect on the parent minus the average influence of each obseravtion such that X_i belongs to the child in estimating theta of the parent. The point is that I cannot find a reasonable explanation for such a formula. For instance when the influence of the observations in node, say C1, are 0 (i.e. the treatment effect in the parent does not change if estimated with or without such observations), according to (6) the theta of C1 is equal to the theta of the parent. Moreover, the formula implies that the higher the influence of the observations in a node in estimating the theta on the parent and the lower the theta on the node.
Therefore the questions are two: 1) how does the formulation of theta of the child in (6) express the idea of maximizing the difference in the treatment effect at each node typical to generalizwd causal forests? 2) can you, please, give me the intuition/main idea of the formulation (6)? Why should the influence of the observations in the child node in estimating the treatment effect of the parent, be subtracted to the treatment effect of the parent (theta_P) to determine theta_C?
Thank you,
Federico
