Describe the workflow you want to enable

Hi, while working on a classification problem I found out there is no dedicated function to compute the P4 metric implemented in sklearn. As a reminder, P4 metrics is a binary classification metric that is commonly seen as an extension of the f_beta metrics because it takes into account all four True Positive, False Positive, True Negative and False Negative values, and because is it symmetrical unlike the f_beta metrics.

P4 is defined as follows : P4 = 4 / ( 1/precision + 1/recall + 1/specificity + 1/NPV )

Wikipedia page right here

Medium article right there

Describe your proposed solution

My idea was to create a function p4_support similar to precision_recall_fscore_support. Since it is a binary metric, multiclass and multi-label inputs would be managed with multilabel_confusion_matrix so the arguments for average would be 'macro', 'samples', 'weighted', 'binary', None.
I would compute all necessaries values such as 1/precision, 1/recall, 1/specificity and 1/NPV using _prf_divide. If any of these four ratios are zero divisions, then P4 would also return the zero division argument. Indeed, for example if precision is null, then 1/precision is +inf and the whole denominator of the P4 is +inf which make P4 = 0 (Btw, this behavior is a reason why it is harder to achieve a high P4 score than f_score since all four ratios need to be 1 to have a P4 equals to 1.). The function would return the tuple (p4_value, support)

A second function p4_score which would be the one actually used by users would return only the first element of the previously described p4_support function.

Describe alternatives you've considered, if relevant

Extras :

Since specificity and NVP are computed anyway, the p4_support function could return the tuple (specificity, NVP, p4_score, support) and then be called specificity_nvp_p4_support. It would then also be possible to add specificity and NVP functions as well using the same scheme as precision or recall.

Responding to #21000 issue, P4 could be added in the classification_report function and would be a good summary of all TP, FP, TN, FN values and their combinations.

Additional context

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