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I'm trying to understand ROC-curve and AUC characteristic for it and found that behaviour of sklearn function roc_auc_score diverges from my understanding:

roc_auc_score([1, 0, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]) # output: 0.5

As far as I undestand, ROC curve can be defined in the following way: $$ROC(x) = \max \{ TPR(\mathcal{C}_t) \mid FPR(\mathcal{C}_t) \leq x \}$$

Where $TPR$ and $FPR$ is true-positive rate and false-positive rate and $\mathcal{C}_t$ is a classifier with fixed threshold value equals to $t$.

In case where all predicted probabilities are equal we have only two different classifiers: one that always predict one class, and one that always predict another class. In this case we have two corner in our ROC-curve: $(0, 0)$ and $(1, 1)$. And by definition above the curve has a shape of corner:

corner shape

But why sklearn draws curve that shapes single line like this?

single line shape

How should I change the definition of curve to get this kind of picture?

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It looks like we simply need to extend our family of classifiers. We are able to do this with a little bit of randomization. So, we can define classifier $\mathcal{C}_t^p$ in the following way: \begin{equation} \mathcal{C}_t^p(x) = \begin{cases} \texttt{+1}&, \texttt{if } C(x) > t\\ \texttt{-1}&, \texttt{if } C(x) < t\\ \texttt{+1 with probability } p \texttt{ and -1 with } 1-p&, \texttt{if } C(x) = t\\ \end{cases} \end{equation}

After this we can simply adjust our definition of ROC-curve:

$$ROC(x) = \max \{ TPR(\mathcal{C}_t^p) \mid p \in [0..1] \wedge FPR(\mathcal{C}_t^p) \leq x \}$$

It perfectly make sense with only single correction that current TPR, FPR are expected values but not deterministic.

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