Fairness Metrics Specifications

Overview

This document provides detailed mathematical specifications for all fairness metrics implemented in ExFairness.

Notation

$Y$: True label
$\hat{Y}$: Predicted label
$A$: Sensitive attribute (e.g., race, gender)
$X$: Feature vector
$P(\cdot)$: Probability
$E[\cdot]$: Expectation

Group Fairness Metrics

1. Demographic Parity (Statistical Parity)

Definition: The probability of a positive prediction should be equal across groups.

$$P(\hat{Y} = 1 | A = 0) = P(\hat{Y} = 1 | A = 1)$$

Disparity Measure: $$\Delta_{DP} = |P(\hat{Y} = 1 | A = 0) - P(\hat{Y} = 1 | A = 1)|$$

When to Use:

When equal representation in positive outcomes is required
Advertising, content recommendation
When base rates can differ between groups

Limitations:

Ignores base rate differences in actual outcomes
May conflict with accuracy if base rates differ
Can be satisfied by a random classifier

Implementation:

def demographic_parity(predictions, sensitive_attr, opts \\ []) do
  threshold = Keyword.get(opts, :threshold, 0.1)

  group_a_mask = Nx.equal(sensitive_attr, 0)
  group_b_mask = Nx.equal(sensitive_attr, 1)

  rate_a = positive_rate(predictions, group_a_mask)
  rate_b = positive_rate(predictions, group_b_mask)

  disparity = Nx.abs(Nx.subtract(rate_a, rate_b)) |> Nx.to_number()

  %{
    group_a_rate: Nx.to_number(rate_a),
    group_b_rate: Nx.to_number(rate_b),
    disparity: disparity,
    passes: disparity <= threshold,
    threshold: threshold
  }
end

2. Equalized Odds

Definition: True positive and false positive rates should be equal across groups.

$$P(\hat{Y} = 1 | Y = 1, A = 0) = P(\hat{Y} = 1 | Y = 1, A = 1)$$ $$P(\hat{Y} = 1 | Y = 0, A = 0) = P(\hat{Y} = 1 | Y = 0, A = 1)$$

Disparity Measures: $$\Delta_{TPR} = |TPR_{A=0} - TPR_{A=1}|$$ $$\Delta_{FPR} = |FPR_{A=0} - FPR_{A=1}|$$

When to Use:

When both false positives and false negatives matter
Criminal justice (both wrongful conviction and wrongful acquittal are serious)
Medical diagnosis (both missed diagnoses and false alarms matter)

Limitations:

Requires ground truth labels
Can be impossible to achieve with demographic parity when base rates differ
May reduce overall accuracy

Implementation:

def equalized_odds(predictions, labels, sensitive_attr, opts \\ []) do
  threshold = Keyword.get(opts, :threshold, 0.1)

  group_a_mask = Nx.equal(sensitive_attr, 0)
  group_b_mask = Nx.equal(sensitive_attr, 1)

  # Group A
  tpr_a = true_positive_rate(predictions, labels, group_a_mask)
  fpr_a = false_positive_rate(predictions, labels, group_a_mask)

  # Group B
  tpr_b = true_positive_rate(predictions, labels, group_b_mask)
  fpr_b = false_positive_rate(predictions, labels, group_b_mask)

  tpr_disparity = Nx.abs(Nx.subtract(tpr_a, tpr_b)) |> Nx.to_number()
  fpr_disparity = Nx.abs(Nx.subtract(fpr_a, fpr_b)) |> Nx.to_number()

  %{
    group_a_tpr: Nx.to_number(tpr_a),
    group_b_tpr: Nx.to_number(tpr_b),
    group_a_fpr: Nx.to_number(fpr_a),
    group_b_fpr: Nx.to_number(fpr_b),
    tpr_disparity: tpr_disparity,
    fpr_disparity: fpr_disparity,
    passes: tpr_disparity <= threshold and fpr_disparity <= threshold
  }
end

3. Equal Opportunity

Definition: True positive rate (recall) should be equal across groups.

$$P(\hat{Y} = 1 | Y = 1, A = 0) = P(\hat{Y} = 1 | Y = 1, A = 1)$$

Disparity Measure: $$\Delta_{EO} = |TPR_{A=0} - TPR_{A=1}|$$

When to Use:

When the cost of false negatives varies by group
Hiring (missing qualified candidates)
College admissions
Opportunity allocation

Limitations:

Only considers true positive rate, ignores false positive rate
Requires ground truth labels
May allow different false positive rates

Implementation:

def equal_opportunity(predictions, labels, sensitive_attr, opts \\ []) do
  threshold = Keyword.get(opts, :threshold, 0.1)

  group_a_mask = Nx.equal(sensitive_attr, 0)
  group_b_mask = Nx.equal(sensitive_attr, 1)

  tpr_a = true_positive_rate(predictions, labels, group_a_mask)
  tpr_b = true_positive_rate(predictions, labels, group_b_mask)

  disparity = Nx.abs(Nx.subtract(tpr_a, tpr_b)) |> Nx.to_number()

  %{
    group_a_tpr: Nx.to_number(tpr_a),
    group_b_tpr: Nx.to_number(tpr_b),
    disparity: disparity,
    passes: disparity <= threshold,
    interpretation: interpret_equal_opportunity(disparity, threshold)
  }
end

4. Predictive Parity (Outcome Test)

Definition: Positive predictive value (precision) should be equal across groups.

$$P(Y = 1 | \hat{Y} = 1, A = 0) = P(Y = 1 | \hat{Y} = 1, A = 1)$$

Disparity Measure: $$\Delta_{PP} = |PPV_{A=0} - PPV_{A=1}|$$

When to Use:

When the meaning of a positive prediction should be consistent
Risk assessment tools
Credit scoring

Limitations:

Can be incompatible with equalized odds when base rates differ
Requires ground truth labels
May allow different selection rates

5. Calibration

Definition: For any predicted probability, actual outcomes should be equal across groups.

$$P(Y = 1 | S(X) = s, A = 0) = P(Y = 1 | S(X) = s, A = 1)$$

where $S(X)$ is the model's score function.

Disparity Measure (per bin): $$\Delta_{Cal}(b) = |P(Y = 1 | S(X) \in bin_b, A = 0) - P(Y = 1 | S(X) \in bin_b, A = 1)|$$

When to Use:

When probability estimates must be interpretable
Medical risk prediction
Weather forecasting
Any application where probabilities guide decisions

Implementation:

def calibration(probabilities, labels, sensitive_attr, opts \\ []) do
  bins = Keyword.get(opts, :bins, 10)
  threshold = Keyword.get(opts, :threshold, 0.1)

  group_a_mask = Nx.equal(sensitive_attr, 0)
  group_b_mask = Nx.equal(sensitive_attr, 1)

  calibration_a = compute_calibration_curve(probabilities, labels, group_a_mask, bins)
  calibration_b = compute_calibration_curve(probabilities, labels, group_b_mask, bins)

  disparities = Enum.zip(calibration_a, calibration_b)
  |> Enum.map(fn {a, b} -> abs(a - b) end)

  max_disparity = Enum.max(disparities)

  %{
    calibration_a: calibration_a,
    calibration_b: calibration_b,
    disparities: disparities,
    max_disparity: max_disparity,
    passes: max_disparity <= threshold
  }
end

Individual Fairness Metrics

6. Individual Fairness (Lipschitz Continuity)

Definition: Similar individuals should receive similar predictions.

$$d(\hat{Y}(x_1), \hat{Y}(x_2)) \leq L \cdot d(x_1, x_2)$$

where $L$ is the Lipschitz constant and $d$ is a distance metric.

Measurement: For a set of similar pairs $(x_i, x_j)$: $$\text{Fairness} = \frac{1}{|P|} \sum_{(i,j) \in P} \mathbb{1}[|f(x_i) - f(x_j)| \leq \epsilon]$$

When to Use:

When individual treatment is important
Personalized recommendations
Custom pricing

Challenges:

Requires defining similarity metric
Computationally expensive for large datasets
Similarity metric may be domain-specific

7. Counterfactual Fairness

Definition: A prediction is counterfactually fair if it is the same in the actual world and in a counterfactual world where the sensitive attribute is different.

$$P(\hat{Y}_{A \leftarrow a}(U) = y | X = x, A = a) = P(\hat{Y}_{A \leftarrow a'}(U) = y | X = x, A = a)$$

When to Use:

When causal understanding is important
Legal compliance (disparate treatment)
High-stakes decisions

Challenges:

Requires causal graph
Unobserved confounders problematic
Computationally intensive

Disparate Impact Measures

80% Rule (4/5ths Rule)

Definition: The selection rate for the protected group should be at least 80% of the selection rate for the reference group.

$$\frac{P(\hat{Y} = 1 | A = 1)}{P(\hat{Y} = 1 | A = 0)} \geq 0.8$$

Legal Context: Used by EEOC in employment discrimination cases.

Implementation:

def disparate_impact(predictions, sensitive_attr) do
  group_a_rate = positive_rate(predictions, Nx.equal(sensitive_attr, 0))
  group_b_rate = positive_rate(predictions, Nx.equal(sensitive_attr, 1))

  ratio = Nx.divide(group_b_rate, group_a_rate) |> Nx.to_number()

  %{
    ratio: ratio,
    passes_80_percent_rule: ratio >= 0.8,
    interpretation: interpret_disparate_impact(ratio)
  }
end

Impossibility Theorems

Chouldechova's Theorem

Statement: If base rates differ between groups ($P(Y=1|A=0) \neq P(Y=1|A=1)$), it is impossible to simultaneously satisfy:

Predictive parity
Equal false positive rates
Equal false negative rates

Implication: Must choose which fairness definition to prioritize.

Kleinberg et al. Theorem

Statement: Except in degenerate cases, it is impossible to simultaneously satisfy:

Calibration
Balance for the positive class (equal TPR)
Balance for the negative class (equal TNR)

Implication: Trade-offs are necessary when base rates differ.

Metric Selection Guide

graph TD
    A[Start] --> B{Ground truth available?}
    B -->|No| C[Demographic Parity]
    B -->|Yes| D{What matters most?}

    D -->|Equal opportunity| E[Equal Opportunity]
    D -->|Both error types| F[Equalized Odds]
    D -->|Prediction meaning| G[Predictive Parity]
    D -->|Probability interpretation| H[Calibration]
    D -->|Individual treatment| I[Individual Fairness]
    D -->|Causal reasoning| J[Counterfactual Fairness]

    C --> K{Legal compliance?}
    K -->|Yes| L[Add 80% rule check]
    K -->|No| M[Use demographic parity]

    E --> N[Check impossibility theorems]
    F --> N
    G --> N
    H --> N

Intersectional Fairness

For multiple sensitive attributes $(A_1, A_2, ..., A_k)$, fairness metrics can be extended:

Additive Intersectionality: Measure fairness for each attribute independently.

Multiplicative Intersectionality: Measure fairness for all combinations of attributes.

Example: Gender × Race creates 4 groups: (Male, White), (Male, Black), (Female, White), (Female, Black)

Confidence Intervals

All metrics should include confidence intervals:

Bootstrap Method:

def demographic_parity_with_ci(predictions, sensitive_attr, opts \\ []) do
  result = demographic_parity(predictions, sensitive_attr, opts)

  # Bootstrap CI
  bootstrap_samples = Keyword.get(opts, :bootstrap_samples, 1000)
  ci_level = Keyword.get(opts, :confidence_level, 0.95)

  bootstrap_disparities = bootstrap(predictions, sensitive_attr, bootstrap_samples, fn p, s ->
    demographic_parity(p, s).disparity
  end)

  ci = percentile_ci(bootstrap_disparities, ci_level)

  Map.put(result, :confidence_interval, ci)
end

References

Hardt, M., Price, E., & Srebro, N. (2016). Equality of opportunity in supervised learning. NeurIPS.
Chouldechova, A. (2017). Fair prediction with disparate impact. Big Data, 5(2), 153-163.
Kleinberg, J., Mullainathan, S., & Raghavan, M. (2016). Inherent trade-offs in the fair determination of risk scores. ITCS.
Dwork, C., Hardt, M., Pitassi, T., Reingold, O., & Zemel, R. (2012). Fairness through awareness. ITCS.
Kusner, M. J., Loftus, J., Russell, C., & Silva, R. (2017). Counterfactual fairness. NeurIPS.

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Fairness Metrics Specifications

Overview

Notation

Group Fairness Metrics

1. Demographic Parity (Statistical Parity)

2. Equalized Odds

3. Equal Opportunity

4. Predictive Parity (Outcome Test)

5. Calibration

Individual Fairness Metrics

6. Individual Fairness (Lipschitz Continuity)

7. Counterfactual Fairness

Disparate Impact Measures

80% Rule (4/5ths Rule)

Impossibility Theorems

Chouldechova's Theorem

Kleinberg et al. Theorem

Metric Selection Guide

Intersectional Fairness

Confidence Intervals

References

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Fairness Metrics Specifications

Overview

Notation

Group Fairness Metrics

1. Demographic Parity (Statistical Parity)

2. Equalized Odds

3. Equal Opportunity

4. Predictive Parity (Outcome Test)

5. Calibration

Individual Fairness Metrics

6. Individual Fairness (Lipschitz Continuity)

7. Counterfactual Fairness

Disparate Impact Measures

80% Rule (4/5ths Rule)

Impossibility Theorems

Chouldechova's Theorem

Kleinberg et al. Theorem

Metric Selection Guide

Intersectional Fairness

Confidence Intervals

References