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We investigate how asymmetrizing an impurity function affects the choice ofoptimal node splits when growing a decision tree for binary classification. Inparticular, we relax the usual axioms of an impurity function and show howskewing an impurity function biases the optimal splits to isolate points of aparticular class when splitting a node. We give a rigorous definition of thisnotion, then give a necessary and sufficient condition for such a bias to hold.We also show that the technique of class weighting is equivalent to applying aspecific transformation to the impurity function, and tie all these notionstogether for a class of impurity functions that includes the entropy and Giniimpurity. We also briefly discuss cost-insensitive impurity functions and givea characterization of such functions.

Asymmetric Impurity Functions, Class Weighting, and Optimal Splits for Binary Classification Trees