What is a confusion matrix, and what do true positives, true negatives, false positives, and false negatives represent?

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A confusion matrix is a table that visualizes the performance of a classification model by comparing its predictions against the actual labels. It organizes predictions into categories based on whether they were correct and what class was predicted.

For binary classification, the confusion matrix is a 2 by 2 table with four entries that represent all possible prediction outcomes.

True Positives occur when the model correctly predicts the positive class. The actual label was positive, and the model predicted positive. In a disease detection scenario, this represents correctly identifying a patient who has the disease.

True Negatives occur when the model correctly predicts the negative class. The actual label was negative, and the model predicted negative. In disease detection, this represents correctly identifying a healthy patient.

False Positives occur when the model incorrectly predicts the positive class. The actual label was negative, but the model predicted positive. This is also called a Type I error or a false alarm. In disease detection, this represents telling a healthy patient they have the disease.

False Negatives occur when the model incorrectly predicts the negative class. The actual label was positive, but the model predicted negative. This is also called a Type II error or a miss. In disease detection, this represents failing to detect the disease in a sick patient.

The confusion matrix layout typically places actual values in rows and predicted values in columns, though conventions can vary. The diagonal from top-left to bottom-right contains correct predictions, while the off-diagonal elements represent errors.

From these four values, you can derive all common classification metrics.

Accuracy equals true positives plus true negatives divided by all predictions. It measures overall correctness.

Precision equals true positives divided by true positives plus false positives. It measures the reliability of positive predictions.

Recall equals true positives divided by true positives plus false negatives. It measures the completeness of positive detection.

Specificity equals true negatives divided by true negatives plus false positives. It measures the ability to correctly identify negatives.

For multi-class classification, the confusion matrix extends to an N by N table where N is the number of classes. Each row represents actual class labels, and each column represents predicted labels. Diagonal entries show correct predictions, while off-diagonal entries reveal which classes are confused with each other.

Examining the confusion matrix provides insights that summary metrics cannot capture. You can identify specific class pairs that the model struggles to distinguish and understand the types of errors your model makes.