A long time ago I seem to remember learning that rather than just assigning a 0 or 1 accuracy values for samples, one could calculate the weighted accuracy of a sample based on the confidence level of a classifier. For example, if p(Y=1|data) = 0.8 and Y=1, then the sample accuracy would be considered to be 0.8, rather than 1. Similarly if Y != 1, then the output is whatever the predicted accuracy of the actual class membership is. So for binary classification, the output would be 0.2.
This approach is then useful for comparing classifiers because the final accuracy increases as confidence of a classifier increases.
My question is, despite being relatively sure that this technique is a thing, I can't find any discussion of it online. Is there a name for this? Weighted accuracy? class-probability accuracy? Jim's old-fashioned log-accuracy? I was hoping to find wording that was a bit more formal than what I had described, as well as any extended discussion relating to use of this approach.