Exercises — Bias, fairness, and discrimination metrics

Demographic Parity. It only asks that the positive-prediction rate be equal across groups: $P(\hat{Y}=1\mid A=0)=P(\hat{Y}=1\mid A=1)=\mathrm{0.45.}✓$ It uses only the prediction $\hat{Y}$ and group $A$ — it never looks at the truth label $Y$. So it says nothing about TPR, FPR, or PPV, all of which depend on $Y$. Equal approval rates can still hide wildly different error rates.

(a) ::: PPV (precision) (b) ::: TPR (recall / true positive rate) (c) ::: FPR (false positive rate) Memory hook: PPV reads across a prediction row; TPR and FPR read down a truth column.

Group A: ${\text{TPR}}_A=\frac{80}{80+20}=0.80,{\text{FPR}}_A=\frac{60}{60+340}=0.15,{\text{PPV}}_A=\frac{80}{80+60}\approx \mathrm{0.5714.}$ Group B: ${\text{TPR}}_B=\frac{150}{150+50}=0.75,{\text{FPR}}_B=\frac{90}{90+210}=0.30,{\text{PPV}}_B=\frac{150}{150+90}=\mathrm{0.625.}$ What we did: for each rate we picked the correct denominator — TPR divides by the truth-positive column ($TP+FN$), FPR by the truth-negative column ($FP+TN$), PPV by the prediction-positive row ($TP+FP$).

Equalized Odds needs both TPR and FPR equal across groups. ${\text{TPR}}_A=0.80\ne 0.75={\text{TPR}}_B,{\text{FPR}}_A=0.15\ne 0.30={\text{FPR}}_B.$ Violated on both counts — Group B is missed less often but falsely flagged twice as often. Errors concentrate against Group B.

Flag (denial) rates: $P(\hat{Y}=1\mid A)=\frac{140}{500}=0.28$ (A), $\frac{240}{500}=0.48$ (B). Because flagging is the adverse outcome, the favourable outcome is "not flagged": $0.72$ (A) and $0.52$ (B). The disadvantaged (protected) group is the one with fewer favourable outcomes — Group B. $\text{DI}=\frac{P(\text{favourable}\mid \text{B})}{P(\text{favourable}\mid \text{A})}=\frac{0.52}{0.72}\approx 0.7222<\mathrm{0.8.}$ Fails the four-fifths rule — the ratio $0.72$ is below the $0.8$ screening threshold. See Disparate impact theory.

Base rate $b=\frac{200}{500}=0.40$, so $\frac{b}{1-b}=\frac{0.40}{0.60}=0.6\overline{6}$. With ${\text{PPV}}_B=0.625$: $\frac{1-0.625}{0.625}=\frac{0.375}{0.625}=0.6$. With ${\text{TPR}}_B=0.75$: ${\text{FPR}}_B=0.6\overline{6}\times 0.6\times 0.75=\mathrm{0.30.}✓$ This matches the directly-computed ${\text{FPR}}_B=0.30$. The identity is not a fairness law — it is an algebraic fact that ties base rate, precision and error rates together, and that is exactly what makes some metrics incompatible.

Rearrange the identity for PPV. From $\text{FPR}=\frac{b}{1-b}\cdot \frac{1-\text{PPV}}{\text{PPV}}\cdot \text{TPR},$ let $k=\frac{b}{1-b}\cdot \frac{\text{TPR}}{\text{FPR}}$. Then $\frac{1-\text{PPV}}{\text{PPV}}=\frac{1}{k}$, giving $\text{PPV}=\frac{k}{k+1}$.

Group A: $\frac{b}{1-b}=\frac{0.5}{0.5}=1$, so $k_A=1\cdot \frac{0.9}{0.2}=4.5$, ${\text{PPV}}_A=\frac{4.5}{5.5}\approx \mathrm{0.8182.}$ Group B: $\frac{b}{1-b}=\frac{0.3}{0.7}\approx 0.4286$, so $k_B=0.4286\cdot 4.5\approx 1.9286$, ${\text{PPV}}_B=\frac{1.9286}{2.9286}\approx \mathrm{0.6585.}$ ${\text{PPV}}_A\approx 0.818\ne 0.659\approx {\text{PPV}}_B.$ Equal error rates + unequal base rates force unequal precision. This is the COMPAS risk assessment controversy in miniature: matching FPR and TPR (ProPublica's demand) automatically breaks calibration/PPV (Northpointe's defence). Both were correct; the metrics are just incompatible. See also Fairness-accuracy tradeoffs.

Required bound: $L\cdot d=6\times 0.02=0.12$. Actual output gap: $|0.81-0.44|=0.37$. $0.37>0.12\Rightarrow \text{Violates individual fairness.}$ Smallest passing $L$: solve $L\cdot 0.02\ge 0.37\Rightarrow L\ge \frac{0.37}{0.02}=18.5$. Why a big $L$ is bad: $L$ is the model's sensitivity — how many units of output swing per unit of input change. Needing $L=18.5$ means a nearly-irrelevant ZIP-code nudge moves the score by 18.5× its size. That is the fingerprint of a hidden proxy (ZIP → neighbourhood → protected attribute). Contrast with Counterfactual fairness: individual fairness bounds observed similarity; counterfactual fairness asks what the score would be in a world where the protected attribute were changed. See Causal fairness and Intersectionality in ML for why a single ZIP proxy can compound across overlapping groups.

Prioritise equal TPR across groups (a slice of Equalized Odds): since missing a sick patient is the catastrophic error, every community must have the same chance of being caught when truly ill. Sacrifice Predictive Parity (equal PPV): by the Chouldechova identity, equalising TPR/FPR under different base rates mathematically forces PPV to differ — so we accept that the meaning of a "positive flag" varies slightly by group in exchange for equal life-saving recall. This is an explicit, documented value choice, the essence of Fairness-accuracy tradeoffs.

I can name which counts sit in each cell of the confusion grid ::: TP top-left, FP top-right, FN bottom-left, TN bottom-right (rows = prediction, columns = truth) I can pick the right denominator for TPR / FPR / PPV without looking ::: TPR ÷ (TP+FN), FPR ÷ (FP+TN), PPV ÷ (TP+FP) I can state why Equalized Odds and Predictive Parity clash ::: the Chouldechova identity fixes PPV once base rate, TPR, FPR are set; unequal base rates force unequal PPV I can flip Disparate Impact correctly when the flag is the bad outcome ::: use the favourable (not-flagged) rate, disadvantaged group over advantaged group