I would say that there might not be any particular or only one measure which you should take into account.
Last time when I did probabilistic classification I had a R package ROCR and explicit cost values for the False Positives and False Negatives.
I considered all cutoff-points from 0 to 1 and used many measures such as expected cost when selecting this cutoff - point. Of course I had already AUC measure for the general measure of classifying accuracy. But for me this was not the only possibility.
Values for the FP and FN cases must come outside your particular model, maybe these are provided by some subject matter expert?
For example in customer churn analysis it might be more expensive to incorrectly infer that customer is not churning but also that it will be expensive to give a general reduction in prices for services without accurary to target these to correct groups.
-Analyst
Introduction
Many practical applications have only positive and unlabeled data (aka PU learning), which poses problems in building and evaluating classifiers. Evaluating classifiers using only positive and unlabeled data is a tricky task, and can only be done by making some assumptions, which may or may not be reasonable for a real problem.
Shameless self-advertisement: For a detailed overview, I suggest reading my paper on the subject.
I will describe the main effects of the PU learning setting on performance metrics that are based on contingency tables. A contingency table relates the predicted labels to the true labels:
+---------------------+---------------------+---------------------+
| | positive true label | negative true label |
+---------------------+---------------------+---------------------+
| positive prediction | true positive | false positive |
| negative prediction | false negative | true negative |
+---------------------+---------------------+---------------------+
The problem in PU learning is that we don't know the true labels, which affects all cells in the contingency table (not just the last column!). It is impossible to make claims about the effect of the PU learning setting on performance metrics without making additional assumptions. For example, if your known positives are biased you can't make any reliable inference (this is common!).
Treating the unlabeled set as negative
A common simplification used in PU learning is to treat the unlabeled set as if it is negative, and then compute metrics as if the problem is fully supervised. Sometimes this is good enough, but this can be detrimental in many cases. I highly recommend against it.
Effect on precision. Say we want to compute precision:
$$p = \frac{TP}{TP + FP}.$$
Now, suppose we have a perfect classifier if we would know the true labels (i.e., no false positives, $p=1$). In the PU learning setting, using the approximation that the unlabeled set is negative, only a fraction of (in reality) true positives are marked as such, while the rest will be considered false positives, immediately yielding $\hat{p} < 1$. Obviously this is wrong, but it gets worse: the estimation error can be arbitrarily large, depending on the fraction of known positives over latent positives. Suppose only 1% of positives are known, and the rest is in the unlabeled set, then (still with a perfect classifier), we would get $\hat{p} = 0.01$ ... Yuck!
Effect on other metrics:
- True Positives: underestimated
- True Negatives: overestimated
- False Positives: overestimated
- False Negatives: underestimated
- Accuracy: depends on balance and classifier
For AUC, sensitivity and specificity I recommend reading the paper as describing it in sufficient detail here would take us too far.
Start from the rank distribution of known positives
A reasonable assumption is that the known positives are a representative subset of all positives (e.g., they are a random, unbiased sample). Under this assumption, the distribution of decision values of known positives can be used as a proxy for the distribution of decision values of all positives (and hence also associated ranks). This assumption enables us to compute strict bounds on all entries of the contingency table, which then translates into (guaranteed!) bounds on all derived performance metrics.
A crucial observation we've made is that in the PU learning context under the assumption mentioned above is that the bounds on most performance metrics are a function of the fraction of positives in the unlabeled set ($\beta$). We have shown that computing (bounds on) performance metrics without an estimate of $\beta$ is basically impossible, as the bounds are then no longer strict.
Best Answer
I do not think this metric has an "official" name. For instance, it does not appear on the very comprehensive Wikipedia page on sensitivity and specificity, which also discusses many other related metrics.
I am a bit doubtful whether your proposed measure is really very useful. Suppose you have a completely random distribution of instances, with >50% positives and <50% negatives. Then you can maximize your criterion $f$ by classifying everything as "positive" - regardless of whether there are 51% true positives, or 60%, or 99.99%. Similarly, if there are <50% positives and >50% negatives, then you will maximize your function by classifying everything as "negative", again regardless of the actual prevalences. This incentive structure does not look very helpful to. (And of course, the same argument holds if these prevalences are conditional on predictors.)
This is very much related to the problems of straight-up accuracy as an evaluation measure, where the exact same problem comes up. I would recommend that you take a look at Why is accuracy not the best measure for assessing classification models? and think about how this applies to your measure.