I have a model which generates probabilities. I pick an optimal threshold for the purposes of classification, by choosing the probability cutoff so that the percentage of cases misclassified is minimised.
I am wondering about the relationship between this threshold and the splitting rule used in a classification tree with one split, such that the root node is split using these probabilities.
If the splitting rule is gini diversity index, the split point of the tree is not the optimal threshold. I have verified that using the optimal threshold for the split point actually results in less information gain.
Why is this happening? Would the split point be the optimal threshold if misclassification rate was used as the splitting rule instead?