## Diagram and Scatter Plot: Endogenous Protected Attributes ### Overview The image presents two distinct visual elements: a directed acyclic graph (DAG) illustrating relationships between endogenous protected attributes, and a scatter plot comparing the causal effect (ATE) against the error rate (1 - AUC) for "Unfair" and "FairPFN" models. ### Components/Axes **Left: Directed Acyclic Graph (DAG)** * **Title:** Endogenous Protected Attribute * **Nodes:** * A1 (light blue circle) * A0 (dark blue circle) * Xf (yellow circle) * Yb (orange circle) * εA0 (green circle) * εYb (green circle) * **Edges:** * A1 -> A0 (solid arrow) * A1 -> Yb (solid arrow) * Xf -> Yb (solid arrow) * A0 -> Yb (solid arrow) * A0 -> εA0 (dashed arrow) * Yb -> εYb (dashed arrow) **Right: Scatter Plot** * **Title:** Endogenous Prot. Attrs. * **X-axis:** Causal Effect (ATE) * Scale: 0.0 to 0.4, incrementing by 0.1 * **Y-axis:** Error (1 - AUC) * Scale: 0.0 to 0.7, incrementing by 0.1 * **Legend (top-right):** * Pink circle: Unfair * Blue star: FairPFN ### Detailed Analysis **Directed Acyclic Graph (DAG)** The DAG depicts causal relationships between variables. A1 and Xf directly influence Yb. A1 also influences A0, which in turn influences Yb. The dashed arrows indicate error terms associated with A0 and Yb. **Scatter Plot** The scatter plot visualizes the relationship between causal effect and error rate for two models: "Unfair" (pink circles) and "FairPFN" (blue stars). * **Unfair (Pink Circles):** The pink circles are scattered across the plot, with a higher concentration in the region of Causal Effect (ATE) between 0.1 and 0.4, and Error (1 - AUC) between 0.1 and 0.5. * **FairPFN (Blue Stars):** The blue stars are more densely clustered in the lower-left region of the plot, indicating lower causal effect and lower error rates compared to the "Unfair" model. The majority of the blue stars are located in the region of Causal Effect (ATE) between 0.0 and 0.2, and Error (1 - AUC) between 0.1 and 0.4. ### Key Observations * The DAG illustrates a causal model with direct and indirect influences between variables. * The scatter plot suggests that the "FairPFN" model generally achieves lower error rates and lower causal effects compared to the "Unfair" model. * There is a significant overlap between the two models, especially in the region of lower causal effect and error. ### Interpretation The image presents a comparison between two models, "Unfair" and "FairPFN," in terms of their causal effect and error rate. The DAG provides a visual representation of the relationships between the variables involved. The scatter plot suggests that the "FairPFN" model is more effective at reducing error, but it also tends to have a lower causal effect. This could indicate a trade-off between fairness and predictive accuracy. The clustering of "FairPFN" points in the lower-left region suggests that this model is generally more desirable in scenarios where both low error and low causal effect are important. The spread of "Unfair" points indicates a wider range of performance, with some instances exhibiting high error and high causal effect.

## Diagram: Causal Diagram & Scatter Plot - Fairness Evaluation ### Overview The image presents two distinct but related elements: a causal diagram illustrating the relationship between protected attributes, features, and outcomes, and a scatter plot comparing the performance of "Unfair" and "FairPFN" models based on causal effect and error. ### Components/Axes **Causal Diagram:** * **Nodes:** A₁ (light blue), A₀ (blue), X_f (yellow), Y_b (orange). * **Edges:** Arrows indicating causal relationships. * **Error Terms:** ε_A0, ε_Yb (text labels near nodes A₀ and Y_b respectively). * **Title:** "Endogenous Protected Attribute" (top-left). **Scatter Plot:** * **X-axis:** "Causal Effect (ATE)" ranging from approximately 0.0 to 0.4. * **Y-axis:** "Error (1 - AUC)" ranging from approximately 0.05 to 0.7. * **Legend:** * "Unfair" (pink circles) * "FairPFN" (blue stars) * **Title:** "Endogenous Prot. Attrs." (top-center). * **Grid:** A light gray grid is present in the background. ### Detailed Analysis or Content Details **Causal Diagram:** The diagram shows A₁ influencing A₀, and X_f influencing Y_b. Both A₀ and X_f influence Y_b. The error terms ε_A0 and ε_Yb represent unmodeled influences on A₀ and Y_b, respectively. **Scatter Plot:** The scatter plot displays the relationship between Causal Effect (ATE) and Error (1 - AUC) for two model types. * **Unfair (Pink Circles):** The pink circles are densely clustered in the top-left corner, with a general downward trend as the Causal Effect increases. * Approximately 20-30% of the points have a Causal Effect between 0.0 and 0.1, with Error values ranging from 0.3 to 0.6. * As the Causal Effect increases to around 0.2, the Error generally decreases, with values ranging from 0.15 to 0.4. * At a Causal Effect of approximately 0.3, the Error values are mostly below 0.2. * There are a few outliers with high Error values (above 0.5) even at low Causal Effect values. * **FairPFN (Blue Stars):** The blue stars are more dispersed than the pink circles. * The stars generally exhibit lower Error values for a given Causal Effect compared to the pink circles. * Approximately 10-15 stars have a Causal Effect between 0.0 and 0.1, with Error values ranging from 0.1 to 0.3. * As the Causal Effect increases to around 0.2, the Error values are generally below 0.2. * At a Causal Effect of approximately 0.3, the Error values are mostly below 0.1. * There are a few outliers with higher Error values (around 0.4) at higher Causal Effect values. ### Key Observations * The "FairPFN" model consistently demonstrates lower Error values for a given Causal Effect compared to the "Unfair" model. * The "Unfair" model exhibits a strong negative correlation between Causal Effect and Error, suggesting that increasing the Causal Effect leads to a reduction in Error. * Both models have outliers, indicating that there are instances where their performance deviates significantly from the general trend. * The scatter plot suggests a trade-off between Causal Effect and Error, with the "FairPFN" model offering a better balance between the two. ### Interpretation The causal diagram illustrates a scenario where a protected attribute (A₁) influences an observed attribute (A₀), which in turn affects the outcome (Y_b) along with other features (X_f). The error terms indicate that the model doesn't capture all the factors influencing these variables. This setup is prone to unfairness if the model doesn't account for the causal relationships. The scatter plot demonstrates the effectiveness of the "FairPFN" model in mitigating unfairness. By achieving lower Error values for a given Causal Effect, the "FairPFN" model suggests a better trade-off between predictive accuracy and fairness. The "Unfair" model, while potentially achieving lower Error values at higher Causal Effects, may be doing so at the cost of exacerbating unfairness. The outliers in both models suggest that there are specific instances where the models struggle to generalize, potentially due to unmodeled factors or data limitations. The diagram and plot together suggest that incorporating causal reasoning into model design (as done in FairPFN) can lead to fairer and more robust predictions, especially in scenarios where protected attributes have an endogenous influence on the outcome. The plot provides empirical evidence supporting the claim that FairPFN reduces error while maintaining a reasonable causal effect.

## Composite Technical Figure: Causal Diagram and Scatter Plot ### Overview The image is a composite figure containing two distinct but related elements. On the left is a causal directed acyclic graph (DAG) illustrating a model for an "Endogenous Protected Attribute." On the right is a scatter plot titled "Endogenous Prot. Attrs." (likely an abbreviation for "Protected Attributes") that compares the performance of two methods, "Unfair" and "FairPFN," across two metrics: Causal Effect (ATE) and Error (1 - AUC). ### Components/Axes **Left Component: Causal Diagram** * **Title:** "Endogenous Protected Attribute" (top-left, italicized). * **Nodes (Variables):** * `A1`: Light blue circle, positioned top-left. * `A0`: Dark blue circle, positioned top-center. * `Xf`: Yellow circle, positioned bottom-left. * `Yb`: Orange circle, positioned bottom-center. * `ε_A0` (epsilon A0): Dark green circle, positioned top-right, connected to `A0` with a dashed line. * `ε_Yb` (epsilon Yb): Dark green circle, positioned bottom-right, connected to `Yb` with a dashed line. * **Edges (Causal Relationships):** Solid black arrows indicate direct influence. * `A1` → `A0` * `A1` → `Yb` * `A0` → `Yb` * `Xf` → `Yb` * Dashed lines connect error terms (`ε_A0`, `ε_Yb`) to their respective variables (`A0`, `Yb`). **Right Component: Scatter Plot** * **Title:** "Endogenous Prot. Attrs." (top-center). * **X-Axis:** * **Label:** "Causal Effect (ATE)" (bottom-center). ATE likely stands for Average Treatment Effect. * **Scale:** Linear, ranging from 0.0 to 0.45. Major ticks at 0.0, 0.1, 0.2, 0.3, 0.4. * **Y-Axis:** * **Label:** "Error (1 - AUC)" (left-center, rotated vertically). AUC likely stands for Area Under the Curve (ROC). * **Scale:** Linear, ranging from 0.05 to 0.7. Major ticks at 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7. * **Legend:** Located in the top-right corner of the plot area. * **Unfair:** Represented by pink/magenta circles (●). * **FairPFN:** Represented by blue stars (★). * **Grid:** Light gray dashed grid lines are present for both major x and y ticks. ### Detailed Analysis **Causal Diagram Analysis:** The diagram models a system where a protected attribute (`A1`) has an endogenous component (`A0`). `A1` influences both the endogenous protected attribute `A0` and the outcome `Yb`. The final outcome `Yb` is also influenced by `A0` and a feature `Xf`. The error terms (`ε_A0`, `ε_Yb`) represent unobserved confounding or noise affecting `A0` and `Yb`, respectively. **Scatter Plot Data Analysis:** * **Data Density:** The plot contains several hundred data points. The "FairPFN" (blue stars) points are densely clustered, while the "Unfair" (pink circles) points are more dispersed. * **"FairPFN" (Blue Stars) Trend & Distribution:** * **Visual Trend:** The cluster slopes gently downward from left to right. * **X-Range (Causal Effect):** Primarily concentrated between 0.0 and 0.2. The vast majority of points are below 0.15. * **Y-Range (Error):** Primarily concentrated between 0.1 and 0.4. The dense core is between 0.15 and 0.35. * **Key Observation:** This method achieves low causal effect (low unfairness) while maintaining moderate to low error. * **"Unfair" (Pink Circles) Trend & Distribution:** * **Visual Trend:** The points are widely scattered with no single clear linear trend, but they occupy a much larger area of the plot. * **X-Range (Causal Effect):** Spans almost the entire axis, from near 0.0 to over 0.4. * **Y-Range (Error):** Also spans a wide range, from below 0.1 to nearly 0.7. * **Key Observation:** This baseline method shows a strong trade-off: points with very low error often have high causal effect (high unfairness), and points with low causal effect often have higher error. There are many outliers with both high error (>0.5) and high causal effect (>0.2). ### Key Observations 1. **Clear Performance Separation:** The two methods form largely distinct clusters. "FairPFN" is tightly grouped in the desirable region of low error and low causal effect. 2. **Trade-off Visualization:** The "Unfair" method's scatter visually demonstrates the fairness-accuracy trade-off. The "FairPFN" cluster appears to break this trade-off, achieving a better Pareto frontier. 3. **Outliers:** Several "Unfair" data points are significant outliers, with Error (1-AUC) values approaching 0.7 and Causal Effect (ATE) values exceeding 0.4. The "FairPFN" method has very few points outside its core cluster. 4. **Spatial Grounding:** The legend is positioned in the top-right, overlapping some of the "Unfair" data points. The highest density of "FairPFN" points is in the center-left of the plot (ATE ~0.05-0.1, Error ~0.2-0.3). ### Interpretation This figure presents a technical argument for a method called "FairPFN" in the context of algorithmic fairness. * **The Causal Model (Left)** defines the problem: it illustrates a scenario where a protected attribute (`A1`) influences an outcome (`Yb`) both directly and through an endogenous component (`A0`), with unobserved factors (`ε`) adding complexity. This setup is typical for studying unfairness where the protected attribute is correlated with other features in the data-generating process. * **The Empirical Results (Right)** demonstrate the solution. The scatter plot provides strong visual evidence that "FairPFN" successfully mitigates the unfairness (low Causal Effect/ATE) without a significant sacrifice in predictive performance (low Error/1-AUC). In contrast, the "Unfair" baseline exhibits the classic, undesirable trade-off: reducing error often increases unfairness, and vice-versa. * **Underlying Message:** The composite figure argues that by explicitly modeling the endogenous nature of protected attributes (as shown in the DAG), the "FairPFN" method can achieve a superior fairness-accuracy balance compared to a standard ("Unfair") approach. The tight clustering of "FairPFN" suggests it is a robust and consistent method across the tested scenarios.

## Diagram: Endogenous Protected Attribute and Scatter Plot of Fairness vs. Causal Effect ### Overview The image contains two components: 1. **Left Diagram**: A causal flow diagram titled "Endogenous Protected Attribute" with colored nodes and directional arrows. 2. **Right Scatter Plot**: A graph titled "Endogenous Prot. Attrs." showing the relationship between "Error (1 - AUC)" and "Causal Effect (ATE)" with two data series differentiated by color and shape. --- ### Components/Axes #### Left Diagram (Causal Flow) - **Nodes**: - **A₁**: Light blue circle (top-left). - **A₀**: Dark blue circle (center-left). - **X_f**: Orange circle (bottom-left). - **Y_b**: Red circle (bottom-center). - **ε_A₀**: Green circle (top-right, connected to A₀). - **ε_Y_b**: Green circle (bottom-right, connected to Y_b). - **Arrows**: - A₁ → A₀ (solid black). - A₁ → Y_b (solid black). - A₀ → Y_b (solid black). - A₀ → ε_A₀ (dashed black). - Y_b → ε_Y_b (dashed black). #### Right Scatter Plot - **Axes**: - **X-axis**: "Causal Effect (ATE)" (0.0 to 0.4). - **Y-axis**: "Error (1 - AUC)" (0.1 to 0.7). - **Legend**: - **Pink Circles**: Labeled "Unfair". - **Blue Stars**: Labeled "FairPFN". - **Data Points**: - Pink circles (Unfair) dominate the upper-left quadrant (high error, low ATE). - Blue stars (FairPFN) cluster in the lower-right quadrant (low error, moderate ATE). --- ### Detailed Analysis #### Left Diagram - **Flow Structure**: - A₁ influences both A₀ and Y_b directly. - A₀ further influences Y_b, creating a dependency chain. - ε_A₀ and ε_Y_b represent exogenous noise terms affecting A₀ and Y_b, respectively. - **Color Coding**: - Blue (A₁, A₀) suggests primary causal variables. - Orange (X_f) and red (Y_b) indicate intermediate/dependent variables. - Green (ε terms) denotes noise. #### Right Scatter Plot - **Data Distribution**: - **Unfair (Pink Circles)**: - Clustered between ATE = 0.0–0.2 and Error = 0.3–0.7. - Outliers extend to ATE = 0.4 (Error ≈ 0.2). - **FairPFN (Blue Stars)**: - Concentrated between ATE = 0.1–0.3 and Error = 0.1–0.3. - Fewer points in the lower-left quadrant (low ATE, low error). --- ### Key Observations 1. **FairPFN vs. Unfair**: - FairPFN methods achieve lower error (1 - AUC) while maintaining moderate causal effect (ATE). - Unfair methods exhibit higher error, especially at lower ATE values. 2. **Causal Flow**: - The diagram suggests Y_b is a downstream variable influenced by both A₀ and A₁, with noise terms ε_A₀ and ε_Y_b introducing variability. 3. **Scatter Plot Trends**: - No clear linear relationship between ATE and Error; FairPFN points show a trade-off between fairness and causal effect. --- ### Interpretation 1. **FairPFN Advantage**: - The scatter plot implies FairPFN methods reduce error (improving AUC) without sacrificing causal effect, making them preferable for fairness-aware modeling. 2. **Endogenous Attribute Dynamics**: - The diagram highlights how protected attributes (A₀) and their noise (ε_A₀) propagate through the system, affecting outcomes (Y_b). This underscores the need to model endogenous confounding in fairness interventions. 3. **Outliers**: - A few Unfair points at high ATE (0.3–0.4) with low error suggest rare cases where unfair methods perform well, possibly due to specific data distributions or model configurations. --- ### Conclusion The image demonstrates that FairPFN methods outperform Unfair approaches in balancing fairness (lower error) and causal effect (ATE). The causal diagram emphasizes the importance of addressing endogenous protected attributes and their noise in fairness-aware machine learning systems.