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I have trouble understanding how nested cross-validation works - I understand the need for two loops (one for selecting the model, and another for training the selected model), but why are they nested?

From what I understood, we need to select the model before training it, which points toward non-nested loops.

Could someone please explain what's wrong (or right?) with my line of reasoning, and also explain nested cross-validation in greater detail? A representative example would be great.

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"Selecting the model" in this case refers to selecting the hyperparameters of the model. The reason to use a nested CV is simply to avoid overfitting training data.

Consider the example in the link. First you like to select the best hyperparameters of your svm model by GridSearchCV(). This is done by 4-fold CV. Now the clf.best_score_ will be the mean cross-validated score of the best estimator (the model with the best hyperparameters). However now you used the same data for training and reporting the performance, although you used CV. Keep in mind that the folds are not independent. Therefore the hyperparameters might be too data specific, i.e. your generalization error estimates are too optimistic. Therefore we like to evaluate our final model performance outside / independent of the hyperparameter selection loop / process (the call cross_val_score()).

In the provided plot, you can clearly see that the reported performance by GridSearchCV() is most of the time better than the performance reported by cross_val_score().

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