# CS 294: Fairness in Machine Learning

## Moritz Hardt September 8, 2017

## Observational fairness criteria

Many definitions

Trade-offs

Algorithms for achieving them

Impossibility results

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## Typical setup

$X$ features of an individual

$A$ sensitive attribute (race, gender, ...)

$C=C(X,A)$ classifier mapping $X$ and $A$ to some prediction

$Y$ actual outcome

Note: random variables in the same probability space

Notation: $\mathbb{P}_a\{E\}=\mathbb{P}\{E\mid A=a\}.$

## All of this is a fragile abstraction

$X$ incorporates all sorts of measurement biases

$A$ often not even known, ill-defined, misreported, inferred

$C$ often not well defined, e.g., large production ML system

$Y$ often poor proxy of actual variable of interest

## Observational criteria

Definition.     A criterion is called observational if it is a property of the joint distribution of features $X,A$, classifier $C$, and outcome $Y$.

Anything you can write down as a probability statement involving $X, A, C, Y.$

## Questions we can ask

What can we learn from observational criteria?

How can we achieve them algorithmically?

How do these criteria trade-off?

How do these criteria shape public discourses?

Key example: COMPAS debate on crime recidivism risk scores

## Demographic parity

Definition.     Classifier $C$ satisfies demographic parity if $C$ is independent of $A$.

When $C$ is binary $0/1$-variables, this means
$\mathbb{P}_a\{C = 1\} = \mathbb{P}_b\{C = 1\}$ for all groups $a,b.$

Approximate versions:

$$\frac{\mathbb{P}_a\{ C = 1 \}} {\mathbb{P}_b\{ C = 1 \}} \ge 1-\epsilon$$
$$\left|\mathbb{P}_a\{ C = 1 \}- \mathbb{P}_b\{ C = 1 \}\right|\le\epsilon$$

## Potential issues

### Lazyness

Accept random people in group $a$ but qualified people in group $b.$

Can happen for lack of training data (recall sample size disparity)

### Not optimality compatible

Rules out perfect classifier $C=Y$ when base rates are different: $\mathbb{P}_a\{C=1\}\ne\mathbb{P}_b\{C=1\}.$

## How about accuracy parity?

Definition.     Classifier $C$ satisfies accuracy parity if
$\mathbb{P}_a\{ C = Y \} = \mathbb{P}_b\{ C = Y \}$ for all groups $a,b$.

## How about accuracy parity?

### The good

Discourages laziness by equalizing error rates across groups.

Allows perfect predictor $C=Y.$

### The bad

Error types matter!

Allows you to make up for rejecting qualified women by accepting unqualified men.

## True positive parity

Assume $C$ and $Y$ are binary $0/1$-variables.

Definition.     Classifier $C$ satisfies true positive parity if
$\mathbb{P}_a\{ C = 1 \mid Y=1\} = \mathbb{P}_b\{ C = 1\mid Y=1\}$ for all groups $a,b$.

Called equal opportunity in H-Price-Srebro (2016).

Suitable when positive outcome ($1$) is desirable.
Equivalently, primary harm is due to false negatives.

## False positive parity

Assume $C$ and $Y$ are binary $0/1$-variables.

Definition.     Classifier $C$ satisfies false positive parity if
$\mathbb{P}_a\{ C = 1 \mid Y=0\} = \mathbb{P}_b\{ C = 1\mid Y=0\}$ for all groups $a,b$.

TPP+FPP together called equalized odds in H-Price-Srebro (2016). We'll also call it positive rate parity.

In full generality, random variable $R$ satisifes equalized odds if $R$ is conditionally independent of $A$ given $Y.$

## Predictive value parity

Assume $C$ and $Y$ are binary $0/1$-variables.

Definition.     Classifier $C$ satisfies
• positive predictive value parity if for all groups $a,b$:
$\mathbb{P}_a\{ Y = 1 \mid C=1\} = \mathbb{P}_b\{Y = 1\mid C=1\}$
• negative predictive value parity if for all groups $a,b$:
$\mathbb{P}_a\{ Y = 1 \mid C=0\} = \mathbb{P}_b\{Y = 1\mid C=0\}$
• predictive value parity if it satisfies both of the above.

## Why predictive value parity?

Equalizes chance of success given acceptance.

## COMPAS: An observational debate ## COMPAS: An observational debate

Probublica's main charge was observational.

Black defendants experienced higher false positive rate.

Northpointe's main defense was observational.

Scores satisfy precision parity.

## Trade-offs are necessary

Proposition.    Assume differing base rates and an imperfect classifier $C\ne Y.$ Then, either
• positive rate parity fails, or
• predictive value parity fails.

Observed in very similar form by Chouldechova (2017).

Similar trade-off result for score functions due to Kleinberg, Mullainathan, Raghavan (2016)

## Intuition

 Group a b Outcome           Unequal base rates Predictor           So far, predictor is perfect.

Let's introduce an error.

## Intuition

 Group a b Outcome           Unequal base rates Predictor           But this doesn't satisfy false positive parity!

Let's fix that!

## Intuition

 Group a b Outcome           Unequal base rates Predictor           Satisfies positive rate parity!

## Intuition

 Group a b Outcome           Unequal base rates Predictor           NPV 2/5 1/3

Does not satisfy predictive value parity!

Proof.    Assume unequal base rates $p_a, a\in\{0, 1\}$, imperfect classifier $C\ne Y$, and positive rate parity. W.l.o.g., $p_0>0$ (since $p_0=p_1=0$ is trivial)
Show that predictive value parity fails.

Proof by googling the first Wiki entry on this:

$\mathrm{PPV_a} = \frac{\mathrm{TPR}p_a}{\mathrm{TPR}p_a+\mathrm{FPR}(1-p_a)}$

Hence, $\mathrm{PPV}_0=\mathrm{PPV}_1$ implies
either $\mathrm{TPR}=0$ or $\mathrm{FPR}=0.$
(But not both, since $C\ne Y$)

$\mathrm{NPV_a} = \frac{(1-\mathrm{FPR})(1-p_a)}{(1-\mathrm{TPR})p_a+(1-\mathrm{FPR})(1-p_a)}$

In either case, $\mathrm{NPV}_0\ne \mathrm{NPV}_1.$ Hence predictive value parity fails.

## Score functions

Formally, any real-valued random variable $R\in[0,1]$ in the same probability space as $(X,A,Y)$.

Leads to family of binary classifiers by thresholding $C=\mathbb{I}\{R>t\}, t\in[0,1]$.

Different thresholds give different trade-offs between true and false positive rate.

## Bayes optimal scores

Goal: Find $R$ that minimizes $\mathbb{E}(Y-R(X,A))^2$

Solution: $R=r(X, A),$ where $r(x, a)=\mathbb{E}[Y\mid X=x, A=a].$

Given score $R$, plot (TPR, FPR) for all possible thresholds   ## Deriving classifiers from scores

Definition.   A classifier $C$ is derived from a score function $R$ if $C=F(R, A)$ where $F$ is a possibly randomized function.

Given score $R$ and cost $c=(c_{fn}, c_{fp}),$
derive classifier $C=F(R, A)$ that minimizes cost

• subject to no constraints.
• subject to equalized odds.
• subject to equality of opportunity.

## Deriving classifiers from scores

Construction on the board. See paper for details.

## Optimality preservation

An optimal equalized odds classifier can be derived
from the Bayes optimal regressor.

Theorem.    Let $R^*=\mathbb{E}[Y\mid X, A]$ be the Bayes optimal unconstrained score and let $c$ be a cost function for true and false positives.
There exists a derived equalized odds classifier $Y^*=F(R^*, A)$ that has minimal cost among all equalized odds classifiers.

Note: Approximate version of the theorem holds as well. See paper.

## FICO credit scores

Based on sample of 301536 TransUnion TransRisk scores from 2003
(made available by US Federal Reserve)

$R$ — Credit score ranging from 300 to 850

$Y$ — Default
Fineprint: failed to pay debt for at least $90$ days on at least one account

$A$ — Race (Asian, Black, Hispanic, White)

## FICO base rates ## FICO score ROC curves by group ## FICO score targets 