Bias, fairness, and discrimination metrics — Hinglish notes

6.4.9 · AI-ML › AI Safety & Alignment

Core Concepts

YEH KYUN MATTER KARTA HAI: Ek statistician ke liye "fair" model (low error rate) ek sociologist ke liye deeply unfair ho sakta hai (groups ke across disparate impact). Hume multiple mathematical definitions chahiye kyunki fairness context-dependent hoti hai.

Perfect Fairness Ki Impossibility

KEY INSIGHT: Zyaadatar fairness metrics mutually exclusive hote hain. Aap sab ko ek saath satisfy nahi kar sakte, sirf trivial cases ko chhodkar.

1. Demographic Parity (Statistical Parity) $P (\hat{Y} = 1∣ A = 0) = P (\hat{Y} = 1∣ A = 1)$ JAHAN:

$\hat{Y}$ : model prediction (1 = positive outcome, e.g., "loan approve karo")
$A$ : protected attribute (0/1, e.g., Group A/B)

YEH DEFINITION KYUN: Groups ke across equal acceptance rates. Positive outcomes mein representation ensure karta hai.

DERIVATION: Fairness principle se shuru karo "positive prediction ka chance group membership par depend nahi karna chahiye." Probability mein translate karo: positive-prediction rate groups ke across same honi chahiye. Formally, kisi bhi do groups $a$ aur $a^{'}$ ke liye: $P (\hat{Y} = 1∣ A = a) = P (\hat{Y} = 1∣ A = a^{'})$ Yeh groups ke across equality hai, marginal rate $P (\hat{Y} = 1)$ ke barabar equality NAHI. (Marginal sirf group rates ka population-weighted average hai; har group ko uske barabar karna sirf tab equivalent statement hai jab exactly do groups hon, lekin ise " $= P (\hat{Y} = 1)$ " ke roop mein likhna misleading hai aur hum isse avoid karte hain.)

2. Equalized Odds (Error Rate Balance) $P (\hat{Y} = 1∣ Y = y, A = 0) = P (\hat{Y} = 1∣ Y = y, A = 1) \forall y \in {0, 1}$

YEH DEFINITION KYUN: Groups ke across equal true positive rates (TPR) AUR equal false positive rates (FPR). Ensure karta hai ki errors ek group mein concentrate na hon.

DERIVATION: Har group ke liye confusion matrix consider karo. Fair treatment ke liye, "jab hum sahi hain" (TPR) aur "jab hum galat hain" (FPR) dono match karne chahiye: \begin{align} \text{TPR}_A &= \frac{TP_A}{TP_A + FN_A} = \text{TPR}_B \ \text{FPR}_A &= \frac{FP_A}{FP_A + TN_A} = \text{FPR}_B \end{align}

3. Predictive Parity (Calibration) $P (Y = 1∣ \hat{Y} = 1, A = 0) = P (Y = 1∣ \hat{Y} = 1, A = 1)$

YEH DEFINITION KYUN: Groups ke across equal precision. Agar model kehta hai "approve karo," toh woh dono groups ke liye equally trustworthy hona chahiye.

DERIVATION: Precision measure karta hai "jinhe humne positive predict kiya, unme se kitne truly positive the?" Fairness ke liye, yeh success rate group-independent honi chahiye: $Precision_{A} = \frac{T P _{A}}{T P _{A} + F P _{A}} = Precision_{B}$

Yeh Metrics Conflict Kyun Karte Hain (The Math)

SETUP: Do groups with different base rates:

Group A: $P (Y = 1∣ A = 0) = 0.5$ (50% actually qualify karte hain)
Group B: $P (Y = 1∣ A = 1) = 0.3$ (30% actually qualify karte hain)

CLAIM: Hum Equalized Odds AUR Predictive Parity dono simultaneously satisfy nahi kar sakte jab base rates alag hon (aur classifier imperfect ho).

PROOF (woh exact identity jo conflict force karti hai): Har group ke liye, precision (PPV), FPR, aur base rate $b = P (Y = 1∣ A = a)$ ek algebraic identity se linked hain. FPR aur TPR ki definition se shuru karo aur law of total probability se $P (\hat{Y} = 1)$ ko expand karo: $P (\hat{Y} = 1∣ A = a) = true positives TPR \cdot b + false positives FPR \cdot (1 - b)$ Precision true positives divided by all positives hai: $PPV = \frac{TPR \cdot b}{TPR \cdot b + FPR \cdot ( 1 - b )}$ PPV, TPR, aur $b$ ke terms mein FPR solve karne par Chouldechova identity milti hai: $FPR = \frac{b}{1 - b} \cdot \frac{1 - PPV}{PPV} \cdot TPR$

YEH KYUN MATTER KARTA HAI: Maano hum Equalized Odds impose karte hain, toh TPR aur FPR dono groups mein equal hain. Tab equation force karti hai: $\frac{b _{A}}{1 - b _{A}} \cdot \frac{1 - PPV _{A}}{PPV _{A}} = \frac{b _{B}}{1 - b _{B}} \cdot \frac{1 - PPV _{B}}{PPV _{B}}$ Kyunki $b_{A} = 0.5 \neq = 0.3 = b_{B}$ , base-rate factors $\frac{b}{1 - b}$ alag hain ( $1.0$ vs $0.43$ ), isliye $PPV_{A} \neq = PPV_{B}$ . Equal error rates isliye unequal precision force karte hain — Predictive Parity violate hoti hai. QED.

(Note: hum perfect classification assume NAHI karte. Ek perfect classifier $\hat{Y} = Y$ banata, jo trivially sab satisfy kar leta lekin base rates alag hone par prediction rates kabhi equalize nahi karta — isliye interesting, real-world case upar wala imperfect classifier hai.)

Practical Metrics & Computation

$Disparate Impact = \frac{P ( Y ^ = 1∣ A = protected )}{P ( Y ^ = 1∣ A = privileged )}$

Legal threshold: Ratio $< 0.8$ discrimination suggest karta hai (U.S. EEOC guideline).

EXAMPLE CALCULATION:

Model Group A applications mein se 60% approve karta hai
Model Group B applications mein se 40% approve karta hai
Disparate Impact = $\frac{0.40}{0.60} = 0.67 < 0.8$ ❌ Fairness test fail

0.8 KYUN? Empirical threshold jo statistical variation vs. systematic bias ko balance karta hai. Sharp boundary nahi hai, lekin ek screening tool hai.

$d (\hat{Y}_{i}, \hat{Y}_{j}) \leq L \cdot d (X_{i}, X_{j})$

JAHAN:

$d (\hat{Y}_{i}, \hat{Y}_{j})$ : predictions mein difference
$d (X_{i}, X_{j})$ : feature space mein distance
$L$ : Lipschitz constant (sensitivity bound)

YEH DEFINITION KYUN: "Similar individuals ko similar predictions milni chahiye." Near-identical applicants ke beech arbitrary discrimination rokta hai.

DERIVATION: "Similar logon ke saath similarly treat karo" ko smoothness constraint ke roop mein formalize karo. Analysis mein Lipschitz continuity se borrow karo: output change, input change se $L$ se scaled hokar bounded hota hai.

EXAMPLE: Do applicants identical credit scores, income, debt ke saath:

Applicant 1: Predicted score 0.72
Applicant 2: Predicted score 0.39
Feature distance: $∣∣ X_{1} - X_{2} ∣∣ = 0.01$ (ZIP code mein tiny difference)

Agar $L = 5$ , required: $∣0.72 - 0.39∣ \leq 5 \cdot 0.01 = 0.05$ . Lekin $0.33 > 0.05$ ❌ Individual fairness violate hoti hai.

YEH STEP KYUN? Tab expose karta hai jab tiny, irrelevant feature changes huge prediction swings cause karein — hidden bias ka red flag.

Group Fairness Measure Karna: Worked Example

SCENARIO: Ek bank ML use karta hai predict karne ke liye ki kaun loan par default karega. Yahan $\hat{Y} = 1$ ka matlab hai "predicted to default" (applicant ke liye ek negative outcome — jinhe flag kiya gaya unhe deny kiya jaata hai). Hum labeled historical data par audit karte hain jahan $Y = 1$ ka matlab hai applicant actually default kiya.

"Approvals" ko "predicted defaults" se confuse na karne ke liye, hum har group ke liye full confusion matrix rakhte hain.

Group A (1000 applicants, 200 actual defaulters):

	Actually defaulted ( $Y = 1$ )	Did not default ( $Y = 0$ )	Row total
Predicted default ( $\hat{Y} = 1$ )	$T P_{A} = 180$	$F P_{A} = 120$	$300$
Predicted no default ( $\hat{Y} = 0$ )	$F N_{A} = 20$	$T N_{A} = 680$	$700$
Column total	$200$	$800$	$1000$

Group B (1000 applicants, 300 actual defaulters):

	Actually defaulted ( $Y = 1$ )	Did not default ( $Y = 0$ )	Row total
Predicted default ( $\hat{Y} = 1$ )	$T P_{B} = 270$	$F P_{B} = 130$	$400$
Predicted no default ( $\hat{Y} = 0$ )	$F N_{B} = 30$	$T N_{B} = 570$	$600$
Column total	$300$	$700$	$1000$

STEP 1: Demographic Parity — flag hone ki rate (predicted default): $P (\hat{Y} = 1∣ A = 0) = \frac{300}{1000} = 0.30, P (\hat{Y} = 1∣ A = 1) = \frac{400}{1000} = 0.40$ YEH ALAG KYUN HAIN? Group B ko 33% zyaada flag kiya jaata hai. Kyunki flagging denial lead karta hai, yeh disparate impact ka red flag hai. (Yeh step kyun? Hum predicted-positives ka row total group size se divide karte hain — true labels ke saath kabhi mix nahi karte.)

STEP 2: Equalized Odds — TPR $Y = 1$ column use karta hai, FPR $Y = 0$ column use karta hai: $TPR_{A} = \frac{T P _{A}}{T P _{A} + F N _{A}} = \frac{180}{200} = 0.90, TPR_{B} = \frac{270}{300} = 0.90 ✓$ $FPR_{A} = \frac{F P _{A}}{F P _{A} + T N _{A}} = \frac{120}{800} = 0.15, FPR_{B} = \frac{130}{700} = 0.186$ YEH KYUN MATTER KARTA HAI: TPRs match karte hain (0.90 = 0.90) lekin FPRs alag hain (0.15 vs 0.186) — Group B ke non-defaulters ko galat zyaada flag kiya jaata hai. Equalized Odds FPR par partially violate hoti hai. (Yeh step kyun? TPR hamesha $T P / (T P + F N)$ hota hai, kabhi row total nahi — pehle ka "impossible TPR > 1" bug galat denominator se divide karne ki wajah se aaya tha.)

STEP 3: Predictive Parity — precision predicted-positive row use karta hai: $Precision_{A} = \frac{T P _{A}}{T P _{A} + F P _{A}} = \frac{180}{300} = 0.60, Precision_{B} = \frac{270}{400} = 0.675$ YEH ALAG KYUN HAIN? Flag kiye logon mein se, Group A ka 60% truly default karta hai vs Group B ka 67.5%. (Yeh step kyun? Precision total predicted positives = $T P + F P$ = row total 300 ya 400 se divide karta hai — approvals ki number ya group size se NAHI.)

CONCLUSION: Demographic Parity fail, Equalized Odds ka FPR-part fail, aur unequal precision dikhata hai. Kyunki base rates alag hain (20% vs 30% default), Chouldechova identity guarantee karti hai ki hum teeno ko ek saath fix nahi kar sakte. Investigate karo:

Base-rate difference (Group B zyaada often default karta hai) — kya yeh real hai ya proxy artifact?
Feature quality (Group A ke liye missing data?)
Labeled dataset mein aane wale logon mein historical bias.

Common Mistakes & Fixes

YEH SAHI KYUN LAGTA HAI: Agar dataset mein race nahi hai, toh model race ke basis par discriminate nahi kar sakta.

YEH GALAT KYUN HAI: Proxy variables (ZIP code, naam, school) race se correlate karte hain. Model indirect discrimination seekh leta hai.

THE FIX:

Protected groups par fairness metrics measure karo chahe training mein use na kiye hon
Adversarial debiasing use karo: ek secondary model train karo jo aapke model ki hidden layers se protected attribute predict kare. Agar woh succeed kare toh penalize karo (model ko group information "bhoolne" ke liye force karta hai).

FORMULA: $L = L_{task} - λ L_{adversary}$ JAHAN $L_{adversary}$ hidden representations se $A$ predict karne ki koshish karta hai.

YEH SAHI KYUN LAGTA HAI: Har kisi ke liye same error rate.

YEH GALAT KYUN HAI: Accuracy TPR aur TNR ko mix karti hai. Aap wildly different error types ke saath equal accuracy rakh sakte ho. Example:

Group A: 90% accuracy (95% TPR, 85% TNR)
Group B: 90% accuracy (70% TPR, 99% TNR)

Group B ko kam false alarms milte hain lekin zyaada true positives miss hote hain. Agar outcome medical diagnosis hai, Group B ka cancer undetected reh jaata hai.

THE FIX: Group ke hisaab se disaggregated metrics (TPR, FPR, precision, recall) report karo, sirf accuracy nahi.

YEH SAHI KYUN LAGTA HAI: $T P$ ko group ke total se divide karna feel karata hai jaise "sabme se humne kitne sahi pakde."

YEH GALAT KYUN HAI: Woh precision nahi hai — woh positive detection rate weirdly scaled hai. Precision ko predicted positives ke row total ( $T P + F P$ ) se divide karna zaroori hai. Ise approvals ya group size se confuse karne par nonsense milta hai (jaisa ki impossible TPR $> 1$ bug jahan $270/300$ galti se galat denominator ke upar rate ke roop mein likha gaya tha).

THE FIX: Har metric ko confusion matrix ke sahi slice se anchor karo:

TPR $= T P / (T P + F N)$ → $Y = 1$ column se divide karo
FPR $= F P / (F P + T N)$ → $Y = 0$ column se divide karo
Precision $= T P / (T P + F P)$ → $\hat{Y} = 1$ row se divide karo

Mitigation Strategies

Representation balance karne ke liye instance weights assign karo: $w_{i} = \frac{P ( A = a ) P ( Y = y )}{P ( A = a , Y = y )}$

KYUN: Underrepresented (group, label) pairs ko upweight karo. Model ko minority cases par dhyan dene ke liye force karta hai.

DERIVATION: Hum chahte hain training $A$ aur $Y$ ko independent "dekhe." Agar Group B with $Y = 1$ rarely appear kare, toh har aisi instance ko zyaada weight do taaki weighted joint, marginals ke product ke barabar ho: $w_{i} \cdot P (A = a, Y = y) = P (A = a) P (Y = y)$

Fairness constraints satisfy karne ke liye per group alag decision thresholds choose karo: $\hat{Y}_{a} = 1 [\overset{p}{^} (X) > τ_{a}]$

EXAMPLE:

Equalized Odds require karo
Group A: $τ_{A} = 0.5$ set karo TPR=0.85 paane ke liye
Group B: $τ_{B} = 0.4$ set karo taaki TPR=0.85 bhi mile

LEGAL KYUN HAI? Agar base rate differences se justified ho aur historical bias perpetuate na kare.

Kab Kaun Sa Metric Use Karein

Metric	Use Case	Example
Demographic Parity	Opportunities mein equal representation	College admissions
Equalized Odds	Equal error rates (justice system)	Bail decisions, recidivism
Predictive Parity	Positive predictions mein trust	Medical diagnosis
Individual Fairness	Arbitrary local discrimination rokna	Insurance quotes

CONTEXT KYUN MATTER KARTA HAI: "Sahi" metric harm model par depend karta hai. Cancer screening mein false negatives ≠ spam detection mein false negatives.

Recall Feynman Explanation (ELI12)

Socho tum aur tumhara dost dono school soccer team ke liye apply kar rahe ho. Tum dono same speed se daudte ho, equally well kick karte ho. Lekin coach ka ek "scoring system" hai jo height ke liye extra points deta hai. Tum chhote ho, tumhara dost lamba hai. Kya yeh fair hai? Demographic Parity kehta hai: "Chhote aur lambe bacchon ka same percentage team mein aana chahiye." Sunta hai fair, lekin kya agar lamba hona sach mein better goalkeeper banata hai?

Equalized Odds kehta hai: "Jin bacchon mein great player banne ka talent hai (truth), coach ko unhe same fraction mein — chaahe chhote hon ya lambe — pick karna chahiye. Jin bacchon mein average talent hai (truth), coach ko dono groups mein galti se same fraction pick karna chahiye." Yeh aise hai jaise keh rahe ho "dono groups ke liye same types ki mistakes karo."

Predictive Parity kehta hai: "Jab coach keh raha hai 'tum star banoge,' toh woh chhote aur lambe bacchon ke liye equally accurate hona chahiye." Agar coach lambe bacchon ke liye 80% sahi hai lekin chhote bacchon ke liye sirf 50%, toh yeh unfair hai — chhote bacchon ko false hope ya unfair doubt milta hai.

Wild part? Teeno ko ek saath satisfy nahi kar sakte agar chhote aur lambe bacchon ki actual skill distributions alag hon. Math prove karta hai. Toh hume choose karna padta hai ki har situation ke liye konsi fairness sabse zyaada matter karti hai.

Connections

Algorithmic accountability: Fairness metrics accountability frameworks ke liye quantitative foundation hain
Disparate impact theory: Legal concept jo 80% rule ke roop mein formalize kiya gaya
Counterfactual fairness: Poochta hai "kya outcome change hota agar protected attribute flip ho jaata?"
Intersectionality in ML: Protected attributes ke combinations ke across fairness (race × gender)
COMPAS risk assessment: Real-world case study jo Predictive Parity violate karta hai
Fairness-accuracy tradeoffs: Dono achieve karne ki theoretical limits
Causal fairness: Legitimate vs. discriminatory paths distinguish karne ke liye causal graphs use karna

#flashcards/ai-ml

80% rule (disparate impact) kya hai :: Ek practical fairness threshold: protected group ke liye selection rate / privileged group ke liye selection rate ≥ 0.8. 0.8 se neeche discrimination suggest karta hai (U.S. EEOC guideline).

Demographic Parity ko mathematically define karo :: $P (\hat{Y} = 1∣ A = 0) = P (\hat{Y} = 1∣ A = 1)$ . Groups ke across equal positive-prediction rates (marginal rate ke barabar nahi), ground truth se regardless.

Equalized Odds define karo :: $P (\hat{Y} = 1∣ Y = y, A = 0) = P (\hat{Y} = 1∣ Y = y, A = 1)$ for both $y = 0$ and $y = 1$ . Groups ke across equal TPR aur FPR.

Predictive Parity define karo :: $P (Y = 1∣ \hat{Y} = 1, A = 0) = P (Y = 1∣ \hat{Y} = 1, A = 1)$ . Groups ke across equal precision (PPV).

Equalized Odds aur Predictive Parity ko conflict karne ke liye kaun si identity force karti hai? :: Chouldechova identity $FPR = \frac{b}{1 - b} \cdot \frac{1 - PPV}{PPV} \cdot TPR$ . Agar base rates $b$ alag hon lekin TPR aur FPR equalize ho jaayein, toh PPV alag hona hi padega.

Individual Fairness (Lipschitz condition) kya hai :: $d (\hat{Y}_{i}, \hat{Y}_{j}) \leq L \cdot d (X_{i}, X_{j})$ . Similar individuals (feature space mein) ko similar predictions milte hain. Input perturbations ke liye output sensitivity bound karta hai.

ML mein proxy variables kya hain :: Wo features jo protected attributes se correlate karte hain (e.g., ZIP code ↔ race). Training se protected attribute hata dene se discrimination nahi rukti agar proxies bachi rahen.

ML bias ke teen sources batao :: (1) Historical bias (training data past discrimination reflect karta hai), (2) Representation bias (undersampled groups), (3) Measurement bias (label/feature quality groups ke beech vary karti hai).

Confusion matrix se precision (PPV) kaise compute karte hain? :: $Precision = T P / (T P + F P)$ — true positives ko total predicted positives ( $\hat{Y} = 1$ row) se divide karo, kabhi group size ya approvals ki number se nahi.