Confusion Matrix

the fundamental counting table for binary classification, splitting samples into 4 cells by “true label × predicted label”: TP (predicted positive, actually positive), FP (predicted positive, actually negative — Type I error), FN (predicted negative, actually positive — Type II error), TN (predicted negative, actually negative), with total $N = TP + FP + FN + TN$. Accuracy $Acc = \frac{TP + TN}{N}$ is the only metric determined solely by the diagonal. Numerical example: 100 samples with 95 negatives and 5 positives; the model predicts everything as negative, giving $TP=0, FP=0, FN=5, TN=95$ and $Acc = 95/100 = 95\%$ — it looks excellent, yet not a single positive is found (recall = 0). This is the accuracy trap.

the starting point of every classification evaluation; all derived metrics (Precision/Recall/F1, specificity, ROC) are read off these 4 cells; for multiclass it generalizes to an $N \times N$ matrix (diagonal = correct predictions), from which macro/micro/weighted averaging is derived.

accuracy is kidnapped by the majority class on imbalanced data (a model that always predicts negative still scores 99% for a 1% prevalence disease), and it is blind to the asymmetric costs of FP vs FN; the confusion matrix exposes both error types and their scale. Macro averaging weights each class equally, $Macro\,P = \frac{1}{C}\sum_{c=1}^{C}P_c$, so rare classes count as much as common ones; micro averaging pools all samples first, $Micro\,P = \frac{\sum_c TP_c}{\sum_c (TP_c + FP_c)}$, and is dominated by the largest class — choose macro when classes are severely imbalanced and the rare class matters, micro when class sizes are balanced or only overall performance matters.

Hand computation: given $TP=40, FP=10, FN=20, TN=30$, compute accuracy, precision, recall and F1.

Why is a model that always predicts negative, at 99% accuracy for a 1% prevalence disease, useless? Which cell gets sacrificed?

Difference between Type I (FP) and Type II (FN) errors; which cell is false-blocking vs missing spam?

How macro vs micro averaging are computed and differ; why choose macro when classes are severely imbalanced (e.g. 1000:1)?

How to represent and read a multiclass confusion matrix (diagonal, row-normalized, weighted average)?