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There are common ways to split a tree in decision trees and all their variants:

  • Gini Index
  • Entropy
  • Misclassification

Why there is not a method which uses directly AUC or accuracy (or whichever the modeler need) to split the nodes.

Is it because of common use, or there is a mathematical explanation for it?

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On accuracy:
Why we use information gain over accuracy as splitting criterion in decision tree?

AUC has been explored; it seems to work well, but is slower:
https://www.semanticscholar.org/paper/Learning-Decision-Trees-Using-the-Area-Under-the-Ferri-Flach/46e40f487e555277033f188778d6c5c05df8daa4
http://proceedings.mlr.press/v7/doetsch09.html

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