Let’s see some examples.

import pandas as pd
d = pd.DataFrame({'prob' : [0.9,0.8,0.8,0.7,0.7,0.5,0.4,0.3,0.2,0.1], 'label': [1,1,1,1,1,0,0,0,0,0]})
d

##    prob  label
## 0   0.9      1
## 1   0.8      1
## 2   0.8      1
## 3   0.7      1
## 4   0.7      1
## 5   0.5      0
## 6   0.4      0
## 7   0.3      0
## 8   0.2      0
## 9   0.1      0

The above can be thought of as a perfect classifier as it fully separates positive from negative instances at 0.5. The perfect AU PR curve is as follows:

import matplotlib.pyplot as plt
from sklearn.metrics import PrecisionRecallDisplay, roc_curve, RocCurveDisplay, auc
display = PrecisionRecallDisplay.from_predictions(d.label, d.prob, name="Perfect classifier, AUPR")     

Similarly, the perfect AUC ROC curve:

def get_auc_plot(d,t):
    fpr, tpr, thresholds = roc_curve(d.label, d.prob)
    roc_auc = auc(fpr, tpr)
    display = RocCurveDisplay(fpr=fpr, tpr=tpr, roc_auc=roc_auc,
                                   estimator_name=t)
    display.plot()
get_auc_plot(d,"Perfect classifier, AUC ROC")

Now let’s generate a random classifier (i.e., all instances have the same probability of being positive):

# Random: same output every time
r = d.copy()
#np.random.seed(0)
r['prob'] = [0.2 for _ in range(10)] # np.random.randint(2, size=10)
display = PrecisionRecallDisplay.from_predictions(r.label, r.prob, name="Random classifier, AUPR")    

By default, the above functions adds two points: (0,1) and (0, positive ratio in dataset). If we don’t want the second point, we can draw the AU PR curve as follows:

display.plot(drawstyle="default")

## <sklearn.metrics._plot.precision_recall_curve.PrecisionRecallDisplay object at 0x309460670>

For the same random classifier, the AUC ROC is:

get_auc_plot(r, "Random classifier, AUC ROC")

An explanation of why AUC ROC is overly optimistic

Consider a dataset that has 10 positives and 100K negatives. Compare two models:

• Model A: Predicts 900 positives, 9 of which are TP
• Model B: Predicts 90 positives, 9 of which are TP

Obviously Model B is better. However:

• Model A: TPR = 9/10, FPR = 81/100K
• Model B: TPR= 9/10, FPR = 891/100K

Because the denominator is so big for FPR, these two models will end up having very similar AUC curves (FPR will be close to zero). Simply put, a large change in the number of false positives resulted in a tiny change in the FPR and thereby, ROC is unable to reflect the superior performance of Model B in the context that true negatives are not relevant to the problem.

In contrast, the Precision-Recall (PR) curve is specifically tailored for the detection of rare events and is the metric that should be used when the positive class is of more interest than the negative one. Because precision and recall don’t consider true negatives, the PR curve is not affected by the data imbalance. Back to the example above:

• Model A: recall = TPR = 0.9 and precision = 9/900 = 0.01
• Model B: recall = TPR = 0.9 and precision = 9/90 = 0.1

Clearly, PR analysis is more informative compared to the ROC analysis above.