F1 Score and F-Beta

Precision asks what share of predicted positive cases are truly positive; recall asks what share of actual positive cases were found. F1 combines them as the harmonic mean, 2 × precision × recall divided by their sum when that denominator is nonzero. The harmonic mean falls when one component is low, so a classifier cannot earn a high F1 merely by excelling at one side. The score is useful for comparing systems with the same task definition, but it does not tell a team which kind of mistake matters more. Consider invented counts of eight true positives, two false positives and eight false negatives. Precision is 8/(8+2) = 0.8 and recall is 8/(8+8) = 0.5. F1 is 2 × 0.8 × 0.5/(0.8+0.5), about 0.615. The equivalent count formula is 2TP/(2TP+FP+FN), or 16/26. Report the counts as well as the score so readers understand what happened. F-beta generalizes F1. When beta is greater than one, recall carries more weight; when beta is less than one, precision does. F2 may suit a screening workflow where missing a relevant case is more costly than sending another case for review, but the choice needs an operational justification. It does not eliminate the need to inspect false positives, capacity limits or subgroup outcomes. A threshold change can trade precision against recall and therefore alter F-beta even if the underlying score model is unchanged. For multiple classes, macro averaging computes the per-class result and then gives each class equal weight. Micro averaging pools true positives, false positives and false negatives across classes before computing the score. These answer different questions when class frequencies vary. State the positive label and averaging rule explicitly. An F1 number says nothing directly about probability calibration, economic cost, or whether a small subgroup is being missed. Evaluate those separately, and use a held-out dataset that represents the decisions the model will face.