## Density Plot: Adult Census Income ### Overview The image presents six density plots, arranged in a 2x3 grid, visualizing the distribution of income-related metrics for different fairness interventions on adult census data. The plots in the top row show the distribution of income (INC), while the bottom row shows the distribution of the absolute difference between income under different conditions. The columns represent different fairness interventions: "Unfair", "Unaware", and "FairPFN". Each plot displays two distributions: "Real" (actual data) and "Cntf." (counterfactual data). ### Components/Axes **Overall Title:** Adult Census Income **Top Row Plots:** * **X-axis:** *IÑC* (Income), ranging from 0.0 to 1.0, with tick marks at intervals of 0.2. * **Y-axis:** Density, ranging from 0 to 7 (Unfair), 0 to 5 (Unaware, FairPFN), with tick marks at intervals of 1. * **Titles:** Unfair, Unaware, FairPFN * **Legend:** Located in the top-right corner of each plot. * "Real": Solid color fill (blue for Unfair, orange for Unaware, pink for FairPFN). * "Cntf.": Dashed line with a lighter shade of the corresponding color. **Bottom Row Plots:** * **X-axis:** |*IÑC*_a→a' - *IÑC*_a→a| (Absolute difference in income), ranging from 0.0 to 0.5, with tick marks at intervals of 0.1. * **Y-axis:** Density, ranging from 0 to 6 (Unfair), 0 to 9 (Unaware), 0 to 13 (FairPFN), with tick marks at intervals of 2. ### Detailed Analysis **Top Row Plots:** * **Unfair (Top-Left):** * **Real (Blue):** The "Real" distribution has two peaks, one around 0.1 and another around 0.4. The density at 0.1 is approximately 3.7, and at 0.4 it is approximately 2.2. * **Cntf. (Dashed Blue):** The "Cntf." distribution has a sharp peak around 0.0, reaching a density of approximately 7.0. It also has a smaller peak around 0.4, with a density of approximately 1.0. * **Unaware (Top-Middle):** * **Real (Orange):** The "Real" distribution has a peak around 0.1, with a density of approximately 3.0, and a broader peak around 0.4, with a density of approximately 2.5. * **Cntf. (Dashed Orange):** The "Cntf." distribution has a peak around 0.1, with a density of approximately 5.0. * **FairPFN (Top-Right):** * **Real (Pink):** The "Real" distribution has two peaks, one around 0.1 and another around 0.3. The density at 0.1 is approximately 5.0, and at 0.3 it is approximately 2.5. * **Cntf. (Dashed Pink):** The "Cntf." distribution has two peaks, one around 0.1 and another around 0.3. The density at 0.1 is approximately 5.0, and at 0.3 it is approximately 3.5. **Bottom Row Plots:** * **Unfair (Bottom-Left):** The distribution has two peaks, one around 0.05 and another around 0.25. The density at 0.05 is approximately 5.2, and at 0.25 it is approximately 2.5. * **Unaware (Bottom-Middle):** The distribution has two peaks, one around 0.05 and another around 0.15. The density at 0.05 is approximately 9.0, and at 0.15 it is approximately 3.2. * **FairPFN (Bottom-Right):** The distribution has a single sharp peak around 0.05, with a density of approximately 12.5. ### Key Observations * The "Cntf." distributions in the top row tend to have a higher density near 0.0 compared to the "Real" distributions, especially for the "Unfair" intervention. * The "FairPFN" intervention results in a bottom row distribution that is highly concentrated around 0.0, indicating a smaller difference in income under different conditions. * The "Unaware" intervention shows a higher density in the bottom row plots compared to the "Unfair" intervention, suggesting a larger difference in income under different conditions. ### Interpretation The plots compare the income distributions under different fairness interventions. The top row shows the distribution of income itself, while the bottom row shows the distribution of the absolute difference in income under different conditions. The "Unfair" intervention represents the baseline scenario without any fairness considerations. The "Unaware" intervention likely represents a scenario where fairness is not explicitly addressed in the model. The "FairPFN" intervention represents a scenario where a fairness-aware algorithm (likely based on counterfactual reasoning) is used. The data suggests that the "FairPFN" intervention is more effective in reducing the difference in income under different conditions, as evidenced by the high concentration of the distribution around 0.0 in the bottom-right plot. This indicates that the algorithm is successful in mitigating unfairness by making the income more consistent across different conditions. The "Unaware" intervention, on the other hand, appears to exacerbate the difference in income, as shown by the higher density in the bottom-middle plot. The "Unfair" intervention falls somewhere in between, with a moderate difference in income. The shift in the "Cntf." distributions towards 0.0 in the top row plots, particularly for the "Unfair" intervention, suggests that the counterfactual reasoning is identifying and correcting for biases that lead to lower income predictions.

## Histograms: Adult Census Income Fairness Evaluation ### Overview The image presents six histograms arranged in a 2x3 grid, evaluating the fairness of different models ("Unfair", "Unaware", "FairPFN") in predicting adult census income. Each model is assessed using two metrics: ÎNC (likely representing an income prediction confidence score) and the absolute difference between ÎNC values under different conditions (ÎNC_α→α' - ÎNC_α→α). Each histogram displays the density distribution of these metrics for both "Real" data and a "Cntf." (likely counterfactual) dataset. ### Components/Axes * **Title:** "Adult Census Income" - positioned at the top-center of the image. * **Subtitles:** "Unfair", "Unaware", "FairPFN" - positioned above each column of histograms. * **X-axis Label (Top Row):** "ÎNC" - appears under each histogram in the top row. Scale ranges from 0.0 to 1.0. * **X-axis Label (Bottom Row):** "|ÎNC_α→α' - ÎNC_α→α|" - appears under each histogram in the bottom row. Scale ranges from 0.0 to 0.5. * **Y-axis Label (All Histograms):** "Density" - appears on the left side of each histogram. Scale ranges from 0.0 to approximately 6.0 (varying slightly between histograms). * **Legend (All Histograms):** * "Real" - represented by a solid fill color (blue for "Unfair", orange for "Unaware", purple for "FairPFN"). * "Cntf." - represented by a dashed fill color (light blue for "Unfair", light orange for "Unaware", light purple for "FairPFN"). * **Gridlines:** Present on all histograms, aiding in value estimation. ### Detailed Analysis or Content Details **Column 1: "Unfair"** * **Top Histogram (ÎNC):** The "Real" data (blue) shows a bimodal distribution with peaks around ÎNC = 0.25 and ÎNC = 0.65. The "Cntf." data (light blue, dashed) shows a similar bimodal distribution, but with lower density overall and peaks around ÎNC = 0.3 and ÎNC = 0.7. * **Bottom Histogram (|ÎNC_α→α' - ÎNC_α→α|):** The "Real" data (blue) is heavily skewed to the left, with a peak around 0.05 and a long tail extending to 0.4. The "Cntf." data (light blue, dashed) is also skewed to the left, with a peak around 0.03 and a tail extending to 0.3. **Column 2: "Unaware"** * **Top Histogram (ÎNC):** The "Real" data (orange) shows a unimodal distribution with a peak around ÎNC = 0.55. The "Cntf." data (light orange, dashed) shows a similar unimodal distribution, but with a broader peak around ÎNC = 0.5. * **Bottom Histogram (|ÎNC_α→α' - ÎNC_α→α|):** The "Real" data (orange) is centered around 0.1, with a peak around 0.08 and a tail extending to 0.4. The "Cntf." data (light orange, dashed) is also centered around 0.1, with a peak around 0.07 and a tail extending to 0.3. **Column 3: "FairPFN"** * **Top Histogram (ÎNC):** The "Real" data (purple) shows a bimodal distribution with peaks around ÎNC = 0.3 and ÎNC = 0.7. The "Cntf." data (light purple, dashed) shows a similar bimodal distribution, but with a broader peak around ÎNC = 0.3 and ÎNC = 0.7. * **Bottom Histogram (|ÎNC_α→α' - ÎNC_α→α|):** The "Real" data (purple) is centered around 0.15, with a peak around 0.1 and a tail extending to 0.5. The "Cntf." data (light purple, dashed) is also centered around 0.15, with a peak around 0.08 and a tail extending to 0.4. ### Key Observations * The "Unfair" model exhibits the most pronounced bimodal distribution in ÎNC, suggesting significant disparities in confidence scores. * The "Unaware" model shows the most concentrated distribution in ÎNC, indicating more consistent confidence scores. * The "FairPFN" model also exhibits a bimodal distribution in ÎNC, but with a wider spread than the "Unfair" model. * The absolute difference metric (|ÎNC_α→α' - ÎNC_α→α|) is generally lower for the "Unaware" model compared to the "Unfair" and "FairPFN" models, suggesting less sensitivity to changes in input conditions. * The "Cntf." data consistently shows lower density than the "Real" data across all models and metrics. ### Interpretation These histograms are evaluating the fairness of different income prediction models. The ÎNC metric likely represents a confidence score assigned by the model to its income prediction. The absolute difference metric quantifies the change in confidence score when a sensitive attribute is altered (represented by α and α'). The "Unfair" model's bimodal distribution in ÎNC suggests that the model assigns significantly different confidence scores to different groups, potentially indicating bias. The "Unaware" model, designed to be oblivious to sensitive attributes, exhibits a more uniform distribution, suggesting less bias. The "FairPFN" model attempts to mitigate bias through a fairness-aware algorithm, and its distribution falls between the "Unfair" and "Unaware" models. The lower absolute difference metric for the "Unaware" model indicates that its predictions are less sensitive to changes in sensitive attributes, which is a desirable property for a fair model. The consistently lower density of the "Cntf." data suggests that the counterfactual data represents less common or more extreme scenarios. The image demonstrates a trade-off between accuracy and fairness. While the "Unaware" model may be fairer, it might sacrifice some predictive accuracy compared to the "Unfair" model. The "FairPFN" model attempts to balance these two objectives.

## Density Plots: Adult Census Income Fairness Metrics ### Overview The image displays a 2x3 grid of kernel density estimation plots analyzing fairness metrics on the "Adult Census Income" dataset. The plots compare the distribution of a metric called `IÑC` (top row) and the absolute difference in `IÑC` between counterfactual and real groups (bottom row) across three different modeling approaches: "Unfair," "Unaware," and "FairPFN." Each subplot contains two distributions: one for "Real" data (solid fill) and one for "Cntf." (Counterfactual) data (dashed outline with lighter fill). ### Components/Axes * **Overall Title:** "Adult Census Income" (centered at the top). * **Column Headers (Top Row):** "Unfair" (left), "Unaware" (center), "FairPFN" (right). * **Y-Axis (All Plots):** Labeled "Density." The scale varies per plot. * **X-Axis (Top Row):** Labeled `IÑC`. Scale ranges from 0.0 to 1.0. * **X-Axis (Bottom Row):** Labeled `|IÑC_{a→a'} - IÑC_{a→a}|`. Scale ranges from 0.0 to 0.5. * **Legends:** Located in the top-right corner of each top-row subplot. * **Unfair (Blue):** "Real" (solid dark blue), "Cntf." (dashed outline, light blue fill). * **Unaware (Orange):** "Real" (solid dark orange), "Cntf." (dashed outline, light orange fill). * **FairPFN (Pink/Magenta):** "Real" (solid magenta), "Cntf." (dashed outline, light pink fill). * **Grid:** Light gray dashed grid lines are present in all subplots. ### Detailed Analysis **Top Row: Distribution of `IÑC`** * **Unfair (Left, Blue):** * **Real Distribution:** Bimodal. Primary peak at `IÑC ≈ 0.1` (Density ~3.8). Secondary, smaller peak at `IÑC ≈ 0.4` (Density ~2.5). Distribution spans roughly 0.0 to 0.6. * **Cntf. Distribution:** Unimodal and sharply peaked. Peak at `IÑC ≈ 0.05` (Density ~7.0). Much narrower spread than the Real distribution, concentrated below 0.2. * **Trend:** The counterfactual distribution is shifted significantly left (lower `IÑC`) and is more concentrated compared to the real data. * **Unaware (Center, Orange):** * **Real Distribution:** Bimodal. Primary peak at `IÑC ≈ 0.1` (Density ~4.5). Secondary peak at `IÑC ≈ 0.4` (Density ~2.6). Similar shape to the "Unfair" Real distribution. * **Cntf. Distribution:** Broader and more complex. Has a peak near `IÑC ≈ 0.1` (Density ~5.2) and a secondary shoulder/peak around `IÑC ≈ 0.25` (Density ~2.8). Overlaps more with the Real distribution than in the "Unfair" case. * **Trend:** The counterfactual distribution is less sharply peaked and shows more overlap with the real data compared to the "Unfair" model. * **FairPFN (Right, Pink):** * **Real Distribution:** Bimodal. Primary peak at `IÑC ≈ 0.1` (Density ~4.8). Secondary peak at `IÑC ≈ 0.35` (Density ~2.6). * **Cntf. Distribution:** Bimodal, closely mirroring the Real distribution. Peaks at `IÑC ≈ 0.1` (Density ~5.2) and `IÑC ≈ 0.3` (Density ~3.5). * **Trend:** The counterfactual and real distributions are very similar in shape and location, indicating high alignment. **Bottom Row: Distribution of `|IÑC_{a→a'} - IÑC_{a→a}|` (Absolute Difference)** * **Unfair (Left, Blue):** * Distribution is bimodal. Primary peak at a difference of `≈ 0.05` (Density ~5.5). Secondary peak at `≈ 0.25` (Density ~2.6). Shows a significant mass of data points with a large difference in `IÑC` between counterfactual and real groups. * **Unaware (Center, Orange):** * Distribution is bimodal. Primary peak at a difference of `≈ 0.05` (Density ~9.0). Secondary peak at `≈ 0.15` (Density ~3.5). The secondary peak is at a lower difference value than in the "Unfair" plot. * **FairPFN (Right, Pink):** * Distribution is strongly unimodal and sharply peaked near zero. Peak at a difference of `≈ 0.02` (Density ~12.5). The distribution decays rapidly, with very little mass beyond a difference of 0.1. * **Trend:** This plot shows the most concentrated distribution near zero, indicating minimal difference between counterfactual and real `IÑC` values. ### Key Observations 1. **Consistent Real Data Shape:** The "Real" data distribution (solid fill) for `IÑC` (top row) is consistently bimodal across all three models, suggesting an inherent structure in the dataset's fairness metric. 2. **Counterfactual Alignment:** The alignment between "Real" and "Cntf." distributions improves dramatically from left to right: "Unfair" (large shift) -> "Unaware" (partial overlap) -> "FairPFN" (high similarity). 3. **Difference Metric Convergence:** The bottom row shows the distribution of the absolute difference metric collapsing toward zero from "Unfair" to "FairPFN." The "FairPFN" model produces a very sharp peak near zero, indicating its counterfactual predictions are very close to the real ones for this metric. 4. **Peak Density Values:** The maximum density values increase in the bottom row from left to right (Unfair: ~5.5, Unaware: ~9.0, FairPFN: ~12.5), reflecting the increasing concentration of the difference metric near zero. ### Interpretation This visualization demonstrates the effectiveness of the "FairPFN" method in achieving fairness on the Adult Census Income dataset, as measured by the `IÑC` metric. * **What the data suggests:** The `IÑC` metric likely measures some form of influence or inconsistency related to protected attributes (denoted by `a`). The "Unfair" model shows a large discrepancy between real and counterfactual `IÑC` values, meaning changing protected attributes would drastically alter the model's behavior. The "Unaware" model reduces this discrepancy but does not eliminate it. The "FairPFN" model successfully aligns the real and counterfactual distributions, meaning the model's behavior is consistent regardless of the protected attribute value. * **How elements relate:** The top row shows the *absolute values* of the metric, while the bottom row shows the *magnitude of change* when flipping the protected attribute. The progression from left to right across columns tells a story of improving fairness: the counterfactual distribution moves to match the real one (top row), and the difference between them shrinks to near zero (bottom row). * **Notable Anomalies/Trends:** The most striking trend is the transformation of the bottom-row distribution from a broad, bimodal shape ("Unfair") to a sharp, zero-centered spike ("FairPFN"). This is a direct visual indicator of reduced unfairness. The bimodality in the "Real" `IÑC` distributions (top row) is an important baseline characteristic of the data that the fairness interventions must account for.

## Chart/Diagram Type: Density Plots of Adult Census Income Metrics ### Overview The image contains six density plots arranged in a 3x2 grid, comparing "Real" and "Counterfactual" (Cntf.) distributions across different fairness metrics ("Unfair," "Unaware," "FairPFN") and an "INC" metric. Each plot uses distinct colors for Real and Cntf. lines, with x-axes representing normalized values (0–1) and y-axes showing density. The bottom row focuses on absolute differences in "ÎNC" values. ### Components/Axes 1. **Top Row Plots**: - **X-axis**: "ÎNC" (0–1) for Unfair, Unaware, FairPFN. - **Y-axis**: Density (0–8). - **Legends**: - **Unfair**: Blue (Real), Gray (Cntf.). - **Unaware**: Orange (Real), Brown (Cntf.). - **FairPFN**: Pink (Real), Purple (Cntf.). - **Positioning**: Legends in top-right corner of each plot. 2. **Bottom Row Plots**: - **X-axis**: Absolute difference "|ÎNC_{a→a'} − ÎNC_{a→a}|" (0–0.5). - **Y-axis**: Density (0–12). - **Colors**: - First plot: Blue (Real). - Second plot: Orange (Real). - Third plot: Pink (Real). ### Detailed Analysis #### Top Row Trends: 1. **Unfair (Top Left)**: - Real (blue) peaks at ~0.2 (density ~5), with a secondary peak at ~0.4 (density ~3). - Cntf. (gray) peaks at ~0.3 (density ~4), with a smaller peak at ~0.5 (density ~2). 2. **Unaware (Top Middle)**: - Real (orange) peaks at ~0.1 (density ~5), with a secondary peak at ~0.4 (density ~3). - Cntf. (brown) peaks at ~0.3 (density ~4), with a smaller peak at ~0.5 (density ~2). 3. **FairPFN (Top Right)**: - Real (pink) peaks at ~0.1 (density ~5), with a secondary peak at ~0.4 (density ~3). - Cntf. (purple) peaks at ~0.3 (density ~4), with a smaller peak at ~0.5 (density ~2). #### Bottom Row Trends: 1. **First INC Plot (Blue)**: - Peaks at ~0.1 (density ~5) and ~0.3 (density ~3). 2. **Second INC Plot (Orange)**: - Single peak at ~0.1 (density ~8). 3. **Third INC Plot (Pink)**: - Sharp peak at ~0.1 (density ~12), tapering off rapidly. ### Key Observations 1. **Peak Consistency**: - Cntf. lines consistently peak near 0.3–0.5 across Unfair, Unaware, and FairPFN plots. - Real lines peak earlier (0.1–0.4) but with varying secondary peaks. 2. **Density Disparities**: - The third INC plot (pink) has the highest density (~12), suggesting a concentrated distribution of absolute differences. 3. **Color-Legend Alignment**: - All legend colors match their respective lines (e.g., blue = Real in Unfair plot). ### Interpretation - **Fairness Metrics**: - The Real and Cntf. distributions differ significantly across metrics. For example, FairPFN’s Real line shows a sharper secondary peak at 0.4 compared to Unfair’s broader distribution. - Cntf. lines (gray, brown, purple) suggest counterfactual scenarios often shift toward higher ÎNC values (0.3–0.5). - **INC Metric**: - The absolute difference plots (bottom row) highlight variability in ÎNC changes. The third plot’s sharp peak at 0.1 indicates a high frequency of small absolute differences, while the second plot’s single peak suggests a more uniform distribution. - **Implications**: - The data may reflect trade-offs between fairness metrics and income-related outcomes. For instance, FairPFN’s distribution could indicate better alignment with desired fairness criteria compared to Unfair or Unaware. - The INC plots emphasize the magnitude of changes in ÎNC, with the third plot’s high density suggesting most differences are minor. - **Anomalies**: - The Unfair plot’s Real line has a bimodal distribution, which may indicate distinct subgroups in the data. - The third INC plot’s extreme density (~12) contrasts with others, warranting further investigation into its underlying causes.