Model evaluation

Definitions from the confusion matrix#

For binary classification with positive class 1:

TP = true positives    (predicted 1, actually 1)
FP = false positives   (predicted 1, actually 0)   -- Type I error
FN = false negatives   (predicted 0, actually 1)   -- Type II error
TN = true negatives    (predicted 0, actually 0)

Accuracy    = (TP + TN) / (TP + TN + FP + FN)
Precision   = TP / (TP + FP)                      also called PPV
Recall      = TP / (TP + FN)                      also sensitivity, TPR
Specificity = TN / (TN + FP)                      also TNR
FPR         = FP / (FP + TN)                      = 1 - Specificity
F_beta      = (1 + beta^2) * P * R / (beta^2 * P + R)

• P, R — precision and recall
• beta — weight on recall relative to precision. beta = 1 gives F1. beta = 2 weights recall four times as heavily

Note the asymmetry that makes rare-event evaluation hard. Precision's denominator depends on the number of predicted positives. Recall's denominator depends on the number of actual positives. Only precision is affected by class prevalence. That is why precision moves when the base rate moves, and recall does not.

The base-rate problem, formally#

By Bayes' theorem, the positive predictive value is:

PPV = ( Sens * Prev ) / ( Sens * Prev + (1 - Spec) * (1 - Prev) )

• Sens — sensitivity (recall), Spec — specificity, Prev — prevalence of the positive class

Take a test with 99 percent sensitivity and 99 percent specificity applied to a condition with prevalence 0.001. Then PPV = (0.99 * 0.001) / (0.99 * 0.001 + 0.01 * 0.999) ≈ 0.090.

A 99-percent-accurate test yields a positive result that is wrong about 91 percent of the time. This is not a modelling failure; it is arithmetic, and it is why screening programmes are always two-stage.

Threshold-free measures#

ROC-AUC is the area under the TPR-versus-FPR curve. It equals the probability that a random positive scores higher than a random negative. That is the Mann-Whitney U statistic. It is invariant to class prevalence, which is both its strength and its trap.

PR-AUC / average precision integrates precision against recall. Its baseline is the positive rate, not 0.5, so it moves with prevalence.

Davis & Goadrich (2006) proved that a curve dominating in ROC space also dominates in PR space, and conversely. The visual impression, however, differs sharply under imbalance. Saito & Rehmsmeier (2015) showed empirically that ROC plots mislead for rare positives. Use PR curves when positives are rare and you care about them.

Average precision = SUM_n ( R_n - R_{n-1} ) * P_n

• P_n, R_n — precision and recall at the n-th threshold

Prefer average_precision_score over the trapezoidal auc(recall, precision), which interpolates optimistically between operating points.

Matthews correlation coefficient#

MCC = ( TP*TN - FP*FN ) / sqrt( (TP+FP)(TP+FN)(TN+FP)(TN+FN) )

MCC is the Pearson correlation between predicted and true binary labels. It ranges from -1 to +1, with 0 meaning chance. Unlike F1 it uses all four cells of the confusion matrix. It goes high only when the model does well on both classes.

Chicco & Jurman (2020) argue MCC should replace F1 and accuracy as the default binary metric. The argument is strong. F1 ignores TN entirely, so a model that handles the negative class badly can still post a good F1.

Calibration#

Discrimination and calibration are distinct properties. A model can rank perfectly (AUC 1.0) while its probabilities are badly wrong.

Brier score = (1/n) * SUM_i ( p_i - y_i )^2

• p_i — predicted probability for example i, y_i — its true binary label

The Brier score is a strictly proper scoring rule, meaning it is uniquely minimised by reporting true probabilities. Log loss is another. Murphy (1973) decomposes Brier into reliability, resolution, and uncertainty.

Guo et al. (2017) showed that modern deep networks are far worse calibrated than the shallower networks of the 1990s, despite higher accuracy. Increased capacity and reduced regularisation push confidence up faster than correctness.

Temperature scaling, a single scalar fitted on validation data, corrects most of it. Platt scaling (1999) and isotonic regression (Zadrozny & Elkan, 2002) are the classical alternatives.

Expected Calibration Error bins predictions by confidence and averages the confidence-accuracy gap. It is sensitive to binning choices, so report the scheme used.

Regression metrics#

MAE  = (1/n) * SUM |y_i - y_hat_i|
MSE  = (1/n) * SUM (y_i - y_hat_i)^2
RMSE = sqrt(MSE)
MAPE = (100/n) * SUM | (y_i - y_hat_i) / y_i |

RMSE is minimised by the conditional mean, MAE by the conditional median. That is the substantive difference; the robustness of MAE to outliers follows from it rather than being a separate fact.

MAPE has two defects that disqualify it more often than people realise. It is undefined at y_i = 0. And it penalises over-prediction more heavily than under-prediction, which biases forecasts downward. Prefer MASE (Hyndman & Koehler, 2006) for scale-free forecast comparison.

Uncertainty in the estimate itself#

A reported metric is a point estimate from a finite sample and carries variance. For accuracy on n test points, the normal-approximation interval is:

SE = sqrt( acc * (1 - acc) / n )

With n = 1000 and accuracy 0.90, SE ≈ 0.0095, so the 95 percent interval spans roughly 0.881 to 0.919. A competitor scoring 0.905 is not meaningfully better. Use the Wilson interval rather than the normal approximation for small n or extreme rates.

To compare two models on the same test set, use McNemar's test (1947) on the discordant pairs. Dietterich (1998) evaluates the common alternatives and recommends it for the single-test-set case. Cross-validated t-tests violate independence, and Nadeau & Bengio (2003) give the variance correction.

Key references#

• Davis, J. & Goadrich, M. (2006). The Relationship Between Precision-Recall and ROC Curves. ICML.
• Saito, T. & Rehmsmeier, M. (2015). The Precision-Recall Plot Is More Informative than the ROC Plot on Imbalanced Datasets. PLoS ONE 10(3).
• Guo, C., Pleiss, G., Sun, Y. & Weinberger, K. (2017). On Calibration of Modern Neural Networks. ICML.
• Chicco, D. & Jurman, G. (2020). The Advantages of the Matthews Correlation Coefficient over F1 and Accuracy. BMC Genomics 21(6).
• Dietterich, T. (1998). Approximate Statistical Tests for Comparing Supervised Classification Learning Algorithms. Neural Computation 10(7).
• Hyndman, R. & Koehler, A. (2006). Another Look at Measures of Forecast Accuracy. Int. J. Forecasting 22(4).
• Murphy, A. (1973). A New Vector Partition of the Probability Score. J. Applied Meteorology 12(4).

Current state#

Single-number leaderboard metrics are increasingly recognised as inadequate. Current practice favours:

• Disaggregated evaluation — report per-subgroup metrics, since aggregate parity can hide large per-group disparities (Buolamwini & Gebru, 2018).
• Behavioural testing — CheckList (Ribeiro et al., 2020) tests capabilities directly rather than scoring a held-out sample.
• Distribution shift — WILDS (Koh et al., 2021) evaluates on deliberately shifted test distributions. In-distribution accuracy is a weak predictor of performance there.
• Documentation — Model Cards (Mitchell et al., 2019) and Datasheets (Gebru et al., 2021). Both standardise reporting of intended use and evaluation conditions.

For generative models the picture is less settled. Perplexity measures fit to a distribution rather than usefulness. Automated LLM-as-judge protocols correlate reasonably with human preference but carry known position and verbosity biases (Zheng et al., 2023). Treat any single generative benchmark number with more scepticism than a classification metric.

What to learn next#

• Train, test and validation splits — protocols that keep these estimates unbiased.
• Overfitting and underfitting — what the train-test gap in these metrics means.
• Classification — decision rules and cost-sensitive thresholds.