## Density Plot: Law School Admissions ### Overview The image presents six density plots arranged in a 2x3 grid, visualizing the distribution of two variables related to law school admissions under three different scenarios: "Unfair," "Unaware," and "FairPFN." Each scenario has two plots: the top plot shows the distribution of FYA, and the bottom plot shows the distribution of the absolute difference between FYA(a->a') and FYA(a->a). The plots compare "Real" data against "Cntf." (counterfactual) data. ### Components/Axes **Overall Title:** Law School Admissions **Top Row Plots:** * **X-axis (all three):** FŶA, ranging from 0.0 to 1.0, with tick marks at intervals of 0.2. * **Y-axis (all three):** Density, ranging from 0 to 5 (Unfair, Unaware) and 0 to 6 (FairPFN), with tick marks at intervals of 1. * **Titles (left to right):** Unfair, Unaware, FairPFN * **Legend (top-right of each plot):** * "Real": Solid filled area. Blue for "Unfair", Orange for "Unaware", and Pink for "FairPFN". * "Cntf.": Dashed line. Light blue for "Unfair", Light orange for "Unaware", and Light pink for "FairPFN". **Bottom Row Plots:** * **X-axis (all three):** |FŶA_a→a' - FŶA_a→a|, ranging from 0.0 to 0.4, with tick marks at intervals of 0.1. * **Y-axis (all three):** Density, ranging from 0 to 10 (Unfair), 0 to 17.5 (Unaware), and 0 to 70 (FairPFN), with tick marks at varying intervals. * **Titles (left to right):** |FŶA_a→a' - FŶA_a→a| for all three. * **Color:** * "Unfair": Blue * "Unaware": Orange * "FairPFN": Pink ### Detailed Analysis **Top Row - Unfair:** * **Real (Blue):** The "Real" distribution is unimodal, peaking around FŶA = 0.6, with a range from approximately 0.2 to 0.8. * **Cntf. (Light Blue Dashed):** The "Cntf." distribution is also unimodal, peaking around FŶA = 0.3, with a range from approximately 0.0 to 0.6. **Top Row - Unaware:** * **Real (Orange):** The "Real" distribution is unimodal, peaking around FŶA = 0.5, with a range from approximately 0.2 to 0.8. * **Cntf. (Light Orange Dashed):** The "Cntf." distribution is unimodal, peaking around FŶA = 0.4, with a range from approximately 0.1 to 0.7. **Top Row - FairPFN:** * **Real (Pink):** The "Real" distribution is unimodal, peaking around FŶA = 0.4, with a range from approximately 0.2 to 0.6. * **Cntf. (Light Pink Dashed):** The "Cntf." distribution is unimodal, peaking around FŶA = 0.4, with a range from approximately 0.2 to 0.6. The "Real" and "Cntf." distributions are very similar. **Bottom Row - Unfair:** * **Blue:** The distribution is unimodal, peaking around |FŶA_a→a' - FŶA_a→a| = 0.3, with a range from approximately 0.1 to 0.4. **Bottom Row - Unaware:** * **Orange:** The distribution is unimodal, peaking sharply around |FŶA_a→a' - FŶA_a→a| = 0.1, with a range from approximately 0.0 to 0.2. **Bottom Row - FairPFN:** * **Pink:** The distribution is highly concentrated and unimodal, peaking sharply around |FŶA_a→a' - FŶA_a→a| = 0.0, with a very narrow range close to 0. ### Key Observations * In the "Unfair" scenario, the "Real" and "Cntf." distributions of FŶA are noticeably different, suggesting a significant impact of unfairness on the outcome. * In the "Unaware" scenario, the "Real" and "Cntf." distributions of FŶA are somewhat similar, but the peak of the "Real" distribution is slightly shifted to the right compared to the "Cntf." distribution. * In the "FairPFN" scenario, the "Real" and "Cntf." distributions of FŶA are almost identical, indicating that the FairPFN model effectively mitigates unfairness. * The distributions of |FŶA_a→a' - FŶA_a→a| show that the "FairPFN" scenario results in the smallest difference between FŶA_a→a' and FŶA_a→a, suggesting that it promotes fairness. ### Interpretation The density plots illustrate the impact of different fairness interventions on the distribution of FŶA and the absolute difference between FŶA_a→a' and FŶA_a→a. The "Unfair" scenario serves as a baseline, showing a clear difference between the "Real" and "Cntf." distributions. The "Unaware" scenario shows a slight improvement in fairness, while the "FairPFN" scenario demonstrates the most significant reduction in unfairness, as evidenced by the near-identical "Real" and "Cntf." distributions and the highly concentrated distribution of |FŶA_a→a' - FŶA_a→a| around 0. This suggests that the FairPFN model is effective in mitigating unfairness in law school admissions.

## Charts: Law School Admissions - Distribution Analysis ### Overview The image presents six density plots comparing "Real" and "Cntf." (likely representing "Counterfactual") distributions for three different scenarios: "Unfair", "Unaware", and "FairPFN". The x-axis in the top row represents "FŶA" (likely a predicted fairness metric), and the x-axis in the bottom row represents the absolute difference between two fairness predictions. The y-axis in all plots represents "Density". ### Components/Axes * **Title:** Law School Admissions * **Subtitles:** Unfair, Unaware, FairPFN (arranged horizontally across the top row) * **X-axis (Top Row):** FŶA, ranging from 0.0 to 1.0, with increments of 0.1. * **X-axis (Bottom Row):** |FŶA_α→α' - FŶA_α→α|, ranging from 0.0 to 0.4, with increments of 0.1. * **Y-axis (All Plots):** Density, ranging from 0.0 to approximately 5.0 (top row) and 8.0 (bottom row). * **Legend:** * Blue: Real * Hatched Blue: Cntf. (in "Unfair" plot) * Orange: Real * Hatched Orange: Cntf. (in "Unaware" plot) * Magenta: Real * Hatched Magenta: Cntf. (in "FairPFN" plot) ### Detailed Analysis **Unfair (Top-Left)** * **Real (Blue):** The distribution is bimodal, with peaks around 0.25 and 0.65. Density reaches a maximum of approximately 4.2 at 0.25 and 3.8 at 0.65. * **Cntf. (Hatched Blue):** The distribution is similar to "Real" but slightly smoother, with peaks around 0.28 and 0.62. Density reaches a maximum of approximately 3.5 at 0.28 and 3.2 at 0.62. * **Trend:** Both distributions exhibit a similar bimodal pattern. **Unaware (Top-Center)** * **Real (Orange):** The distribution is unimodal, peaking around 0.5. Density reaches a maximum of approximately 4.8 at 0.5. * **Cntf. (Hatched Orange):** The distribution is also unimodal, peaking around 0.5, but is more spread out than "Real". Density reaches a maximum of approximately 3.8 at 0.5. * **Trend:** Both distributions are unimodal, centered around 0.5. **FairPFN (Top-Right)** * **Real (Magenta):** The distribution is unimodal, peaking around 0.8. Density reaches a maximum of approximately 5.0 at 0.8. * **Cntf. (Hatched Magenta):** The distribution is also unimodal, peaking around 0.8, but is more spread out than "Real". Density reaches a maximum of approximately 4.0 at 0.8. * **Trend:** Both distributions are unimodal, centered around 0.8. **Unfair (Bottom-Left)** * **Real (Blue):** The distribution is unimodal, peaking around 0.1. Density reaches a maximum of approximately 7.5 at 0.1. * **Trend:** The distribution is heavily skewed towards lower values. **Unaware (Bottom-Center)** * **Real (Orange):** The distribution is unimodal, peaking around 0.15. Density reaches a maximum of approximately 16.5 at 0.15. * **Trend:** The distribution is heavily skewed towards lower values. **FairPFN (Bottom-Right)** * **Real (Magenta):** The distribution is unimodal, peaking around 0.05. Density reaches a maximum of approximately 65 at 0.05. * **Trend:** The distribution is heavily skewed towards lower values, and the density is significantly higher than in the other bottom-row plots. ### Key Observations * The "Unfair" scenario exhibits a bimodal distribution for both "Real" and "Cntf." values of FŶA, suggesting potential instability or multiple equilibrium points. * The "Unaware" and "FairPFN" scenarios show unimodal distributions for FŶA, indicating a more consistent prediction. * The bottom row plots, representing the absolute difference in fairness predictions, show a strong skew towards zero for all scenarios, but particularly for "FairPFN". This suggests that the fairness predictions are relatively consistent under the "FairPFN" approach. * The density values in the bottom-right plot ("FairPFN") are significantly higher than in the other bottom-row plots, indicating a much smaller difference between the two fairness predictions. ### Interpretation The charts compare the distributions of a fairness metric (FŶA) and the difference between two fairness predictions under different admission scenarios. The "Unfair" scenario shows a bimodal distribution, suggesting that fairness is not consistently achieved. The "Unaware" and "FairPFN" scenarios show unimodal distributions, indicating more consistent fairness predictions. The bottom row plots demonstrate that the "FairPFN" approach results in the smallest difference between the two fairness predictions, suggesting it is the most stable and consistent approach to achieving fairness in law school admissions. The high density near zero in the "FairPFN" bottom plot suggests that the counterfactual fairness prediction is very close to the original fairness prediction, indicating a strong degree of robustness. The bimodal distribution in the "Unfair" scenario suggests that the fairness metric is sensitive to initial conditions or other factors, leading to multiple possible outcomes. The differences in the distributions highlight the impact of different fairness interventions on the consistency and stability of fairness predictions.

## Density Plots: Law School Admissions Fairness Analysis ### Overview The image displays a 2x3 grid of density plots titled "Law School Admissions." It compares the distribution of a fairness-related metric, denoted as **FŶA**, across three different modeling approaches ("Unfair," "Unaware," "FairPFN") under two conditions ("Real" and "Cntf."). The bottom row analyzes the absolute change in this metric when an attribute is altered. ### Components/Axes * **Overall Title:** "Law School Admissions" (centered at the top). * **Column Headers (Top Row):** "Unfair" (left), "Unaware" (center), "FairPFN" (right). * **Top Row Plots:** * **X-axis:** Labeled **FŶA**. Scale ranges from 0.0 to 1.0 with major ticks at 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. * **Y-axis:** Labeled **Density**. Scale varies per plot (0-5 for Unfair/Unaware, 0-6 for FairPFN). * **Legend (in each top plot, top-right corner):** * **Real:** Solid fill color (blue for Unfair, orange for Unaware, magenta for FairPFN). * **Cntf.:** Dashed outline with a lighter, semi-transparent fill of the same hue. * **Bottom Row Plots:** * **X-axis:** Labeled **|FŶA_{a→a'} - FŶA_{a→a}|**. Scale ranges from 0.0 to 0.4 with major ticks at 0.0, 0.1, 0.2, 0.3, 0.4. * **Y-axis:** Labeled **Density**. Scale varies significantly per plot (0-10 for Unfair, 0-17.5 for Unaware, 0-70 for FairPFN). * **No legends** are present in the bottom row plots. The color corresponds to the column (blue, orange, magenta). ### Detailed Analysis **Top Row: Distribution of FŶA** * **Unfair (Blue):** The "Real" distribution is bimodal, with a smaller peak near 0.2 and a dominant peak around 0.55. The "Cntf." distribution is unimodal, peaking sharply around 0.25. There is a significant separation between the two distributions. * **Unaware (Orange):** The "Real" and "Cntf." distributions overlap considerably. Both are unimodal, with the "Real" peak slightly to the right (around 0.48) of the "Cntf." peak (around 0.42). * **FairPFN (Magenta):** The "Real" and "Cntf." distributions are nearly identical, showing almost complete overlap. Both form a single, sharp peak centered at approximately 0.4. **Bottom Row: Distribution of |FŶA_{a→a'} - FŶA_{a→a}|** * **Unfair (Blue):** The distribution of the absolute difference is unimodal and symmetric, peaking at approximately 0.28. The density at the peak is ~9.5. * **Unaware (Orange):** The distribution is unimodal and slightly right-skewed, peaking at approximately 0.08. The density at the peak is ~18. * **FairPFN (Magenta):** The distribution is extremely narrow and tall, peaking very close to 0.0 (approximately 0.01). The density at the peak is exceptionally high, reaching ~72. ### Key Observations 1. **Progression of Overlap:** Moving from left to right (Unfair → Unaware → FairPFN), the "Real" and "Cntf." distributions in the top row show increasing overlap, culminating in near-perfect alignment for FairPFN. 2. **Shift in Central Tendency:** The central peak of the "Real" distribution shifts leftward across the columns: from ~0.55 (Unfair) to ~0.48 (Unaware) to ~0.4 (FairPFN). 3. **Magnitude of Change:** The bottom row reveals a dramatic reduction in the magnitude of the metric's change (|FŶA_{a→a'} - FŶA_{a→a}|) across the models. The peak of the distribution moves from ~0.28 (Unfair) to ~0.08 (Unaware) to ~0.01 (FairPFN). 4. **Precision of FairPFN:** The FairPFN model not only aligns the real and counterfactual distributions but also results in an extremely stable metric (very low absolute change), as indicated by the very high, narrow density peak near zero in the bottom-right plot. ### Interpretation This visualization demonstrates the effectiveness of different algorithmic approaches to fairness in a law school admissions context. **FŶA** likely represents a fairness metric (e.g., related to acceptance rates or outcomes across groups). * The **"Unfair"** model shows a large discrepancy between real-world outcomes ("Real") and a counterfactual scenario ("Cntf."), suggesting the model's decisions are highly sensitive to the attribute being altered (e.g., race, gender). The large absolute differences in the bottom plot confirm this high sensitivity. * The **"Unaware"** model, which likely excludes the sensitive attribute from the decision process, reduces but does not eliminate the discrepancy. Some disparity persists, and the model remains moderately sensitive to attribute changes. * The **"FairPFN"** model achieves two key fairness properties: 1. **Counterfactual Fairness:** The near-perfect overlap of "Real" and "Cntf." distributions indicates that, in a counterfactual world where the sensitive attribute is changed, the model's outcomes would be statistically identical. This is a strong fairness guarantee. 2. **Stability/Robustness:** The extremely low absolute difference values (bottom-right plot) show that the model's fairness metric is highly stable and minimally affected by perturbations to the sensitive attribute. In summary, the figure argues that the FairPFN method successfully aligns real and counterfactual outcomes and produces a robustly fair model, outperforming both a standard ("Unfair") and an attribute-ignorant ("Unaware") approach.

## Line Graphs: Law School Admissions Fairness Metrics ### Overview The image contains six comparative density plots analyzing fairness metrics in law school admissions. Three graphs in the top row compare "Real" and "Counterfactual" (Cntf.) distributions for different fairness approaches ("Unfair," "Unaware," "FairPFN"), while three graphs in the bottom row show the absolute differences between these distributions for the same approaches. ### Components/Axes **Top Row Graphs:** - **X-axis**: FYA (Fairness Yield Advantage) scale 0.0–1.0 - **Y-axis**: Density (0–5) - **Legends**: - Blue solid line = Real - Dashed blue line = Counterfactual - **Graph Titles**: - Unfair - Unaware - FairPFN **Bottom Row Graphs:** - **X-axis**: |FYA_{a→a'} - FYA_{a→a}| (absolute difference) scale 0.0–0.4 - **Y-axis**: Density (scales vary: 0–10, 0–17.5, 0–70) - **Legends**: - Solid color = Real (no counterfactual shown) - **Graph Titles**: - FYA - FYA - FYA ### Detailed Analysis **Top Row Trends:** 1. **Unfair**: - Real distribution: Bimodal peaks at ~0.3 and ~0.6 - Counterfactual: Single peak at ~0.4 - Overlap: Significant between 0.2–0.5 range 2. **Unaware**: - Real: Single peak at ~0.5 - Counterfactual: Slightly shifted peak at ~0.45 - Overlap: Strong between 0.3–0.6 range 3. **FairPFN**: - Real: Sharp peak at ~0.5 - Counterfactual: Narrower peak at ~0.45 - Overlap: Minimal (0.4–0.55 range) **Bottom Row Trends:** 1. **FYA**: - Single sharp peak at ~0.25 - Density: ~8 (top row scale) 2. **Unaware**: - Single peak at ~0.15 - Density: ~15 (middle row scale) 3. **FairPFN**: - Extremely narrow peak at ~0.05 - Density: ~70 (bottom row scale) ### Key Observations 1. **Bimodal Reality**: The "Unfair" approach shows two distinct admission groups (low and high FYA), while counterfactual attempts to merge them into a single group. 2. **Peak Shifts**: All counterfactual distributions show systematic shifts toward lower FYA values compared to real distributions. 3. **FairPFN Precision**: The absolute difference graph for FairPFN shows the smallest disparity (peak at 0.05), indicating minimal fairness impact. 4. **Scale Disparity**: Bottom row graphs show FairPFN's difference metric is 10x narrower than Unaware and 20x narrower than Unfair. ### Interpretation The data demonstrates three distinct fairness approaches in law school admissions: 1. **Unfair** creates/perpetuates systemic disparities (bimodal distribution) 2. **Unaware** reduces but doesn't eliminate disparities (peak shift of 0.05) 3. **FairPFN** achieves near-perfect fairness (difference <0.1) The counterfactual distributions consistently underestimate real-world disparities across all approaches. FairPFN's extreme narrowness in the absolute difference graph suggests it successfully neutralizes admission biases, while Unfair's bimodal pattern reveals entrenched inequities. The density scale differences between graphs highlight how FairPFN's impact is most pronounced despite similar FYA ranges.