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If I have an ROC for a single classifier [y(x) in the range 0...1] that is 'worse than random', namely the AUC of the ROC is less than 0.5, would a classifier that reversed the class predictions [y'(x)=1-y(x)] be better than random to the same degree?

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$y'(x)$ means you will work with $ROC' = ROC^{-1}$ (inverse of $ROC$), as all true positive will be falsely negative and vice versa. Therefore, $AUC' = 1 - AUC$ (As ROC is an increasing function and inside a unit square), and your answer is yes.

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You can reverse the prediction as you say, but you may want to determine why this occurred, so the model could be more robust, and easier to believe or explain. Taking the reverse prediction in many cases can be dangerous.

Some of the reasons you have a low AUC (< 0.5):

1) Labels/features are incorrect/reversed

2) Data was not split randomly/correctly between Train and Test, such that the data in train and test have different patterns - more likely in imbalanced data

3) Problems in the training algorithm

4) Possibly overfitting

5) etc

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