Can We Calculate Auc From Binary Classification Without Probabilities

Can We Calculate AUC from Binary Classification Without Probabilities?

The Area Under the ROC Curve (AUC) is a popular metric for evaluating the performance of binary classification models. It represents the probability that the model will rank a randomly chosen positive instance higher than a randomly chosen negative instance. A higher AUC indicates better performance. But can we calculate AUC without the predicted probabilities that are typically used in its calculation? The short answer is no, not directly. Let's explore why.

Understanding AUC Calculation and its reliance on probabilities

The standard method for calculating AUC involves using the predicted probabilities of each instance belonging to the positive class. These probabilities are then used to rank the instances. The ROC curve is constructed by plotting the true positive rate (TPR) against the false positive rate (FPR) at various probability thresholds. The AUC is then the area under this curve. The process inherently relies on the ordering of instances based on their predicted probabilities. Without probabilities, we lack this crucial ranking information.

Why Probabilities are Essential

To understand why probabilities are needed, consider the following:

• Ranking Instances: AUC measures the model's ability to discriminate between positive and negative instances. This discrimination is quantified by the ranking induced by the predicted probabilities. Higher probabilities indicate a higher likelihood of belonging to the positive class, thus influencing the position of the instance in the ranking. Without probabilities, we lack a consistent method to rank the instances, which is fundamental to the AUC calculation.

• Threshold Independence: The ROC curve and AUC are designed to be threshold-independent. This means that the AUC value remains consistent regardless of the specific classification threshold used. The probability outputs allow us to sweep through different thresholds and generate the complete ROC curve. Without probabilities, the concept of varying thresholds and the resulting curve becomes meaningless.

• Underlying Assumption: The calculation of AUC implicitly assumes that the model provides a measure of confidence or likelihood for each instance belonging to the positive class. This is precisely what the predicted probabilities represent. A simple binary classification (0 or 1) lacks this graded confidence measure.

Alternatives and Misconceptions

While we cannot directly calculate AUC without probabilities, there are some related metrics that can be considered, but they don't represent the same information:

• Accuracy: Accuracy measures the overall correctness of the classifications, but it doesn't capture the ranking ability of the model which is crucial for AUC.

• Precision and Recall: These metrics provide information on the model's performance at a specific threshold, but do not offer the overall ranking information captured by the AUC.

Conclusion

In conclusion, the calculation of AUC fundamentally relies on the predicted probabilities of a binary classification model. These probabilities provide the necessary ranking information that allows us to construct the ROC curve and calculate the area beneath it. While other metrics exist to evaluate binary classification performance, they cannot directly replace the AUC, which is specifically designed to assess the model's ability to rank instances based on their likelihood of belonging to the positive class. Therefore, obtaining the predicted probabilities is essential to compute the AUC.