In this image, I have a series of threshold as follows 0.5, 0.8, 0.85, 0.95. Now in this diagram, if I choose the threshold 0.8 because it has roughly 0.7 True Positive Rate and 0.2 False Positive Rate which is a better choice to someone who wants a higher TPR, then what would be my AUC?
Should I have to take the whole curve as AUC? Then what would be my threshold as my threshold starts from 0.5 and goes all the way till 0.95
The AUC-ROC is defined across all the points shown in that graph. Please note that the threshold used to calculate any of the FPR/TPR rates shown is not directly depicted on that graph. Choosing the threshold should directly relate to misclassification costs. In the absence of them, we can use metrics like the Youden Index or pick the threshold that maximise some other metric (e.g. the $F_1$ score or the Matthews correlation coefficient) but that is not a best practice. ROC curves in a way, came to prominence especially because they are not reliant on a single threshold.
CV.SE has some interesting topics on the matter if you want to explore this further: What are some common methods for choosing a discriminating threshold from an ROC curve? and Optimal classifier or optimal threshold for scoring.
