# What is the integral of the False Positive Rate over the False Positive Rate, compared to the AUC?

In machine learning the Area Under the Receiver Operating Characteristic Curve ($$AUC$$) can be illustrated in a plot of the True Positive Rate ($$TPR$$) against the False Positive Rate ($$FPR$$). Formally, it can also be expressed as the expected value of the $$TPR$$, given a uniform distribution of $$FPR$$:

\begin{align} AUC = \int_{0}^{1}TPR\ dFPR \end{align}

This can also be expressed as as a function of $$TPR$$ and a decision threshold $$k$$ (assuming that first, a scoring function estimates the probability of belonging to the positiv class and an indicator function then assigns the class based on a decision threshold $$k$$): \begin{align} \label{eq:auc} AUC = \int_{0}^{1}TPR\ dFPR = \int_{-\infty}^{\infty} TPR_{k}\ f_{o}(k)dk \end{align} where $$f_{o}(k)$$ is the probability that the scoring function $$F(X)$$ produces a score of exactly $$k$$ in the instances that have a negative class label (see Hand 2009).

I am wondering what is the value of integrating the $$FPR$$ over the $$FPR$$:

\begin{align} \int_{-\infty}^{\infty}FPR\ f_{0}(k)dk = \int_{-\infty}^{\infty}FPR\ dFPR \end{align}

Dalessandro et al. (2014) suggest that

\begin{align} \int_{-\infty}^{\infty}FPR\ f_{0}(k)dk = \int_{-\infty}^{\infty}FPR\ dFPR = 0.5 \end{align}

but this is not very intuitive to me and causes contradictory results in my further proceedings.

References:

HAND, D. J. (2009): “Measuring classifier performance: a coherent alternative to the area under the ROC curve”, Machine Learning, 77, 103–123.

DALESSANDRO, B., C. PERLICH, AND T. RAEDER (2014): “Bigger is Better, but at What Cost? Estimating the Economic Value of Incremental Data Assets”, Big data, 2, 87–96.