Performance Metrics for Machine Learning

a model is only as good as the number you judge it by, and most of the classic modelling disasters are really metric disasters — a fraud detector with \(99.9\%\) accuracy that never flags anything, a medical test tuned to a roc curve nobody deployed at the published threshold. 𐃏 this page is the field guide: what each metric measures, what it silently assumes, and which one to reach for when the classes are lopsided, the probabilities matter, or the target is continuous.

the confusion matrix#

every threshold classifier on a binary problem induces four counts. fix the convention: positive is the rare/interesting class.

the zoo of derived metrics is just different marginalisations of this table:

three of these deserve equations. the f-beta family interpolates between precision and recall:

\begin{equation} F_\beta \;=\; \frac{(1+\beta^2)\,\mathrm{precision}\cdot\mathrm{recall}}{\beta^2\,\mathrm{precision} + \mathrm{recall}}, \end{equation}

where \(\beta\) is how many times more you care about recall than precision — \(F_1\) is the balanced case, \(F_2\) favours recall (screening tests), \(F_{0.5}\) favours precision (spam filters). being a harmonic mean, \(F_\beta\) punishes imbalance between the two: you cannot buy a good \(F_1\) with precision alone. 𐃏

matthews correlation coefficient uses all four cells — it is literally the pearson correlation between the predicted and actual binary labels:

\begin{equation} \mathrm{MCC} \;=\; \frac{\mathrm{tp}\cdot\mathrm{tn} - \mathrm{fp}\cdot\mathrm{fn}} {\sqrt{(\mathrm{tp}+\mathrm{fp})(\mathrm{tp}+\mathrm{fn})(\mathrm{tn}+\mathrm{fp})(\mathrm{tn}+\mathrm{fn})}} \in [-1, 1], \end{equation}

with \(0\) for chance-level prediction and \(-1\) for perfect disagreement. of the single-number summaries it is the hardest to fool with imbalance, because every marginal appears in the denominator.

accuracy is the one to distrust first: with \(10\%\) positives, “always predict negative” scores \(0.90\). any accuracy must be compared against the majority-class baseline \(\max(\pi, 1-\pi)\), where \(\pi\) is the prevalence.

roc curves and auc#

a scoring classifier \(\hat p(x)\) plus a sweep of thresholds \(t\) traces the receiver operating characteristic: the curve of \((\mathrm{fpr}(t), \mathrm{tpr}(t))\) as \(t\) slides from \(+\infty\) (bottom-left) to \(-\infty\) (top-right). 𐃏

• the diagonal \(\mathrm{tpr} = \mathrm{fpr}\) is chance: a classifier that scores by coin toss.
• the point \((0, 1)\) is perfection; curves closer to the top-left dominate.
• the roc is invariant under any strictly increasing transform of the scores — it sees only the ranking. calibration is invisible here (see below).

the probabilistic meaning of auc#

the area under the roc curve has an exact interpretation that makes it worth reporting:

\begin{equation} \mathrm{AUC} \;=\; \mathbb{P}\!\left(\hat p(X^+) > \hat p(X^-)\right) + \tfrac{1}{2}\,\mathbb{P}\!\left(\hat p(X^+) = \hat p(X^-)\right), \end{equation}

for an independently drawn positive \(X^+\) and negative \(X^-\): the probability that a random positive outranks a random negative. equivalently, \(\mathrm{AUC}\) is the normalised mann–whitney \(U\) statistic of the two score samples — which is exactly how you compute it in \(O(n \log n)\) via ranks (code below).¹ chance gives \(0.5\); a perfect ranker gives \(1\); anything below \(0.5\) means you wired the sign backwards.

auc summarises ranking quality across all thresholds — which is also its weakness: deployment happens at one threshold, and auc happily averages over operating regions you will never use.

precision–recall curves and class imbalance#

roc’s axes are both class-conditional rates: tpr conditions on positives, fpr on negatives. neither sees the prevalence \(\pi\). precision does:

\begin{equation} \mathrm{precision} \;=\; \frac{\pi \cdot \mathrm{tpr}}{\pi \cdot \mathrm{tpr} + (1-\pi)\,\mathrm{fpr}}, \end{equation}

by bayes’ rule. with \(\pi = 0.001\) and a “great” roc point \((\mathrm{fpr}, \mathrm{tpr}) = (0.01, 0.9)\), precision is \(\approx 8\%\): eleven false alarms per true hit. the roc curve looks superb; the deployed system is a nuisance.

hence the pr curve — precision against recall as the threshold sweeps — for rare-positive problems (fraud, retrieval, rare disease):

• the chance baseline in pr space is the horizontal line at \(\pi\) (a random scorer’s precision at any recall), so the plot shows lift over prevalence directly, where roc’s diagonal hides it.
• the pr curve is not monotone and interpolation between points is nonlinear — summarise with average precision (area under it) rather than trapezoids.
• a curve dominates in roc space if and only if it dominates in pr space (davis & goadrich, 2006), but differences are magnified near low recall in pr space when \(\pi\) is small — exactly the operating region rare-event systems live in.

rule: report roc when classes are roughly balanced or when class-conditional trade-offs are the object; report pr when positives are rare and flagging cost is the object.

calibration and the brier score#

a classifier can rank perfectly and still lie about probabilities. \(\hat p\) is calibrated when

\begin{equation} \mathbb{P}\!\left(Y = 1 \mid \hat p(X) = q\right) \;=\; q \qquad \text{for all } q \in [0,1]: \end{equation}

of all the days the forecaster says “\(70\%\) rain”, it rains on \(70\%\). check it with a reliability diagram — bin predictions by \(\hat p\), plot observed frequency per bin against bin centre; calibrated models hug the diagonal.

the brier score measures probability quality as plain squared error,

\begin{equation} \mathrm{BS} \;=\; \frac{1}{n} \sum_{i=1}^{n} (\hat p_i - y_i)^2, \end{equation}

and admits the murphy decomposition \(\mathrm{BS} = \text{reliability} - \text{resolution} + \text{uncertainty}\): a calibration penalty, minus a reward for confident discrimination, plus the irreducible \(\bar y(1 - \bar y)\) of the base rate. brier (like log loss) is a proper scoring rule: its expectation is uniquely minimised by reporting the true conditional probability — you cannot game it by hedging or exaggerating (gneiting and raftery 2007, jasa 102(477), is the definitive treatment). auc and accuracy are not proper; a model selected on them alone may need post-hoc recalibration (platt scaling, isotonic regression) before its probabilities mean anything.

regression metrics#

the “minimised by” column is the loss-geometry fact that matters (derived properly in loss functions): choosing rmse versus mae is choosing which functional of the conditional distribution you are estimating — the mean or the median. under skewed noise they genuinely differ, and the metric silently picks for you.

• always \(\mathrm{rmse} \ge \mathrm{mae}\) (jensen), with equality only when all errors have equal magnitude; a large gap flags heavy-tailed residuals.
• \(R^2\) compares against the constant-mean baseline: \(R^2 = 0\) means “no better than predicting \(\bar y\)”, and it is negative whenever the model is worse than that. on held-out data this happens more often than people admit. it is also not comparable across datasets with different target variances (Hastie, Trevor and Tibshirani, Robert and Friedman, Jerome, 2009).
• mae’s median-seeking makes it the honest choice when over- and under-prediction cost the same per unit; rmse when large errors are disproportionately costly.

micro versus macro averaging#

with \(K\) classes, per-class precision/recall/f1 must be aggregated, and the two standard recipes answer different questions:

• micro: pool the \(K\) confusion matrices into one (sum tp, fp, fn across classes), then compute the metric. every instance votes equally, so frequent classes dominate. for single-label multiclass prediction, micro-precision = micro-recall = micro-f1 = accuracy — all four collapse.²
• macro: compute the metric per class, then take the unweighted mean over classes. every class votes equally: a 50-sample rare class counts as much as a 50,000-sample common one. macro-f1 is the standard summary when rare-class competence is the point.
• weighted macro: per-class metrics averaged with class-frequency weights — a compromise that mostly behaves like micro.

the gap between micro and macro is itself a diagnostic: micro \(\gg\) macro means the model is coasting on frequent classes.

everything from scratch#

confusion-matrix zoo, rank-based auc, brier, and the regression metrics — on a synthetic problem with \(10\%\) positives where scores are gaussian (\(\mathcal{N}(1,1)\) for positives, \(\mathcal{N}(0,1)\) for negatives), so the true auc is \(\Phi(1/\sqrt{2}) \approx 0.760\). the posterior \(\hat p_{\mathrm{cal}}\) is computed by bayes’ rule (calibrated by construction); \(\hat p_{\mathrm{sig}} = \sigma(s)\) ignores the prevalence (miscalibrated on purpose).

import numpy as np
from math import erf, sqrt

rng = np.random.default_rng(1)

# synthetic scores: 10% positives; score ~ N(1,1) for +, N(0,1) for -
n, prev = 20000, 0.10
y = (rng.uniform(size=n) < prev).astype(int)
s = np.where(y == 1, rng.normal(1.0, 1.0, n), rng.normal(0.0, 1.0, n))

phi = lambda z: np.exp(-z**2 / 2) / np.sqrt(2 * np.pi)
p_cal = prev * phi(s - 1) / (prev * phi(s - 1) + (1 - prev) * phi(s))  # bayes posterior
p_sig = 1 / (1 + np.exp(-s))              # miscalibrated: ignores prevalence

def confusion(y, yhat):
    tp = int(np.sum((yhat == 1) & (y == 1))); fp = int(np.sum((yhat == 1) & (y == 0)))
    fn = int(np.sum((yhat == 0) & (y == 1))); tn = int(np.sum((yhat == 0) & (y == 0)))
    return tp, fp, fn, tn

print("thresh    acc   prec    rec   spec     f1    mcc")
for t in (0.10, 0.25, 0.50):
    tp, fp, fn, tn = confusion(y, (p_cal >= t).astype(int))
    acc  = (tp + tn) / n
    prec = tp / (tp + fp)
    rec  = tp / (tp + fn)
    spec = tn / (tn + fp)
    f1   = 2 * prec * rec / (prec + rec)
    mcc  = (tp * tn - fp * fn) / np.sqrt(
        float(tp + fp) * (tp + fn) * (tn + fp) * (tn + fn))
    print(f"{t:>6.2f} {acc:>6.3f} {prec:>6.3f} {rec:>6.3f} "
          f"{spec:>6.3f} {f1:>6.3f} {mcc:>6.3f}")

def auc_rank(y, p):                        # mann-whitney rank statistic
    order = np.argsort(p)
    ranks = np.empty(n); ranks[order] = np.arange(1, n + 1)
    npos, nneg = y.sum(), n - y.sum()
    return (ranks[y == 1].sum() - npos * (npos + 1) / 2) / (npos * nneg)

auc_theory = 0.5 * (1 + erf(1 / sqrt(2) / sqrt(2)))   # Phi(1/sqrt(2))
print(f"\nauc: posterior = {auc_rank(y, p_cal):.4f}  sigmoid = {auc_rank(y, p_sig):.4f}"
      f"  theory = {auc_theory:.4f}")

print(f"brier: calibrated = {np.mean((p_cal - y)**2):.4f}"
      f"  miscalibrated = {np.mean((p_sig - y)**2):.4f}"
      f"  predict-prevalence = {np.mean((prev - y)**2):.4f}")

# regression metrics from scratch, prediction = true conditional mean
x = rng.uniform(0, 1, 500)
ytrue = 2 * x + rng.normal(0, 0.3, 500)
yhat = 2 * x
rmse = np.sqrt(np.mean((ytrue - yhat) ** 2))
mae  = np.mean(np.abs(ytrue - yhat))
r2   = 1 - np.sum((ytrue - yhat) ** 2) / np.sum((ytrue - ytrue.mean()) ** 2)
print(f"regression: rmse = {rmse:.4f}  mae = {mae:.4f}  r2 = {r2:.4f}")

thresh    acc   prec    rec   spec     f1    mcc
  0.10  0.696  0.208  0.707  0.695  0.322  0.255
  0.25  0.877  0.360  0.265  0.946  0.305  0.243
  0.50  0.899  0.588  0.039  0.997  0.073  0.133

auc: posterior = 0.7646  sigmoid = 0.7646  theory = 0.7602
brier: calibrated = 0.0820  miscalibrated = 0.2752  predict-prevalence = 0.0916
regression: rmse = 0.2934  mae = 0.2367  r2 = 0.7892

four object lessons in one printout:

• the accuracy trap: at threshold \(0.5\) accuracy is \(0.899\) — while recall is \(0.039\). the majority-class baseline is \(0.90\), so this “\(90\%\)-accurate” model is worse than doing nothing by the only metric it advertises. mcc (\(0.13\)) is not fooled.
• thresholds move everything: precision and recall trade off threefold across three thresholds; a metric quoted without its threshold is an unfinished sentence.
• auc is transform-blind: the calibrated posterior and the naive sigmoid score identical auc (\(0.7646\), matching \(\Phi(1/\sqrt 2)\)) because one is a monotone transform of the other.
• brier is not: the calibrated posterior scores \(0.082\), beating even the “predict the prevalence” straw man (\(0.092\)), while the same-auc sigmoid scores \(0.275\). ranking quality and probability quality are different axes; brier sees the second, auc only the first.

choosing, quickly#

• balanced classes, hard decisions: accuracy is fine; mcc if you want insurance.
• imbalanced, flagging costs matter: pr curve + average precision, f-beta at the deployment threshold.
• comparing rankers across thresholds: roc-auc.
• probabilities feed a downstream decision: brier or log loss, plus a reliability diagram; recalibrate if needed.
• regression: rmse when big misses are catastrophic, mae when they are merely proportional; always sanity-check \(R^2\) against zero.
• multiclass with rare classes you care about: macro-f1, and report the micro-macro gap.

and estimate all of them on data the model never trained on — the machinery for that lives in cross validation.

see also#

• cross validation — how to estimate these metrics without flattering yourself
• loss functions — metrics you can differentiate: the training-time counterparts
• the bias-variance decomposition — what sits underneath a test-set rmse
• the no free lunch theorem — why no single metric (or model) rules them all