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When training decision trees, the standard algorithms (e.g. ID3, C4.5, C5.0) use either the gini index or entropy to determine which node to add next. Only once the tree is built, and the ROC curve is being evaluated in comparison to other classification models, are the decision tree's precision and recall evaluated.

Are there any decision tree building algorithms that use the precision and recall, instead of the gini index or entropy, to decide which node to add next?

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There are two terms here: 1- The objective function: this is the function that the algorithm tries to optimize. In the case of DT it is Gini/Entropy, in the case of Linear Regression, it is Gradient Descent, etc.

2- Then it is the metric that you use to evaluate the overall performance of the model after the model is done. Here you use precision, recall, AUC ROC, etc. These are not objective function you can be optimized.

However, you can use recall/precision for model selection. You can modify your hyperparameters and see which one results in a better recall/precision (closer to what you need). Moreover, if you are really concern about one error type, you can change the weight of different errors, or upsample/downsample your data so that the "metrics" you need is achieved.

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