Equalized Odds (Fairness Analysis)¶

Equalized Odds detects shortcuts by measuring True Positive Rate (TPR) and False Positive Rate (FPR) disparities across protected groups.

Based on Hardt et al. (2016), this method checks whether a classifier trained on embeddings satisfies the equalized odds criterion: equal TPR and FPR across all demographic groups.

Large gaps indicate that the embeddings encode group-dependent shortcuts that cause differential performance.

How It Works¶

The Equalized Odds detector:

Trains a lightweight classifier (LogisticRegression by default) on embeddings to predict task labels
Computes per-group metrics: TPR and FPR for each protected group
Calculates gaps: gap = max(rate) - min(rate) across groups
Assesses risk based on gap thresholds

graph TD
    A[Embeddings + Labels + Groups] --> B[Train Classifier]
    B --> C[Predictions]
    C --> D[Per-Group TPR/FPR]
    D --> E[Compute Gaps]
    E --> F{Gap >= Threshold?}
    F -->|Yes| G[Shortcut Detected]
    F -->|No| H[No Shortcut]

Key insight: If TPR or FPR differs significantly across groups, the model treats groups unequally for the same ground truth, suggesting shortcut reliance.

Basic Usage¶

Via the Unified API (Recommended)¶

from shortcut_detect import ShortcutDetector
import numpy as np

# Precomputed embeddings
embeddings = np.load("embeddings.npy")
labels = np.load("labels.npy")  # Binary labels required
group_labels = np.load("groups.npy")  # Required

detector = ShortcutDetector(methods=["equalized_odds"])
detector.fit(embeddings, labels, group_labels=group_labels)

print(detector.summary())

Standalone Usage¶

from shortcut_detect.fairness import EqualizedOddsDetector

detector = EqualizedOddsDetector(
    tpr_gap_threshold=0.1,
    fpr_gap_threshold=0.1,
    min_group_size=10,
)

detector.fit(embeddings, labels, group_labels)

print(f"TPR Gap: {detector.tpr_gap_:.3f}")
print(f"FPR Gap: {detector.fpr_gap_:.3f}")
print(f"Risk Level: {detector.report_.risk_level}")

Parameters¶

Parameter	Type	Default	Description
`estimator`	Estimator	LogisticRegression	Classifier to train on embeddings
`min_group_size`	int	10	Minimum samples per group (smaller groups get NaN)
`tpr_gap_threshold`	float	0.1	Threshold for flagging TPR disparity
`fpr_gap_threshold`	float	0.1	Threshold for flagging FPR disparity

Outputs¶

Report Structure¶

results_["equalized_odds"]["report"] contains an EqualizedOddsReport:

Field	Type	Description
`group_metrics`	dict	Per-group TPR, FPR, support, and confusion matrix counts
`tpr_gap`	float	Max TPR - Min TPR across groups
`fpr_gap`	float	Max FPR - Min FPR across groups
`overall_accuracy`	float	Classifier accuracy on all samples
`risk_level`	str	"low", "moderate", or "high"
`notes`	str	Human-readable interpretation
`reference`	str	"Hardt et al. 2016"

Per-Group Metrics¶

For each group, group_metrics[group_id] contains:

Metric	Description
`tpr`	True Positive Rate = TP / (TP + FN)
`fpr`	False Positive Rate = FP / (FP + TN)
`support`	Number of samples in the group
`tp`, `fp`, `tn`, `fn`	Confusion matrix counts

Interpretation¶

Risk Assessment¶

Risk Level	Condition	Interpretation
Low	Both gaps < threshold	Equalized odds approximately satisfied
Moderate	Any gap >= threshold	Noticeable disparity in TPR or FPR
High	Any gap >= 2x threshold	Large disparity, strong evidence of shortcuts

What the Metrics Mean¶

Pattern	Interpretation
High TPR gap	Model catches positives better for some groups than others
High FPR gap	Model makes more false alarms for some groups
Both gaps high	Systematic unfairness across both error types
Low accuracy + low gaps	Fair but poorly performing model

Rule of thumb

If TPR or FPR differs by more than 10% across groups, the embeddings likely encode shortcuts correlated with group membership.

Example with Synthetic Data¶

from shortcut_detect import ShortcutDetector
import numpy as np

np.random.seed(42)

# Create embeddings where group 0 has stronger signal
n = 500
embeddings = np.random.randn(n, 20)
labels = (embeddings[:, 0] > 0).astype(int)
groups = np.array([0] * 250 + [1] * 250)

# Add shortcut: group 1 has noisier signal
embeddings[250:, 0] += np.random.randn(250) * 2

detector = ShortcutDetector(methods=["equalized_odds"])
detector.fit(embeddings, labels, group_labels=groups)

print(detector.summary())

Expected: Higher TPR/FPR gaps due to differential signal quality across groups.

When to Use Equalized Odds¶

Use Equalized Odds when:

You have binary task labels (required)
You have explicit group labels (demographics, environments)
You want interpretable fairness metrics
You need to report regulatory fairness compliance
You want to measure differential error rates across groups

Don't use Equalized Odds when:

Labels are multi-class (use other methods)
Group labels are unavailable
You want unsupervised detection (use HBAC, Geometric)
Groups are too small (< 10 samples each)

Theory¶

Equalized Odds (Hardt et al., 2016) requires:

$$ P(\hat{Y}=1 | Y=y, A=a) = P(\hat{Y}=1 | Y=y, A=a') $$

For all $y \in {0, 1}$ and protected groups $a, a'$.

This decomposes into two constraints:

Equal TPR: $P(\hat{Y}=1 | Y=1, A=a) = P(\hat{Y}=1 | Y=1, A=a')$
Equal FPR: $P(\hat{Y}=1 | Y=0, A=a) = P(\hat{Y}=1 | Y=0, A=a')$

The detector measures violations as:

$$ \text{TPR Gap} = \max_a \text{TPR}_a - \min_a \text{TPR}_a $$

$$ \text{FPR Gap} = \max_a \text{FPR}_a - \min_a \text{FPR}_a $$