# Precision and Recall: Valid Combination?

I'm looking at a precision-recall curve for a binary classification task. My precision-recall curve intersects the y-axis (precision) at 60% and the x-axis at 15%. So I get 15% precision at 100% recall.

1) Doesn't this mean that my true label occurs 15% of the time?

2) My curve (using sklearn's precision_recall_curve) goes through the point representing 50% precision and 40% recall. But doesn't that imply that there are 20% positive labels? So isn't that an invalid combination of precision and recall values?

Thanks!

## 1 Answer

1) By "crossing the x-axis", it seems like you're actually meaning where the plot ends when recall=100%. In that case, yes.

(Crossing x-axis means when y=0. But y = precision in this case and precision can't be smaller than proportion of positive samples.)

2) Why does that imply there are 20% positive labels?

For example, if we have the following confusion matrix values (for some threshold): $$\text{TP}=6,\quad \text{FN}=9,\quad \text{FP}=6,\quad\text{TN}=79,$$ then we get precision=50% and recall=40%, and the proportion of positives 15%.

You can get those confusion matrix numbers by doing simple algebra with the following: $$\frac{\text{TP}+\text{FN}}{\text{TP}+\text{FN}+\text{FP}+\text{TN}} = 0,$$ $$\text{precision}=\frac{\text{TP}}{\text{TP}+\text{FP}} = 0.5,$$ $$\text{recall}=\frac{\text{TP}}{\text{TP}+\text{FN}} = 0.4.$$ I used the smallest integer that would satisfy these equations.

I suggest this post to learn more about interpreting Precision-Recall curves!