## Causal Diagram: Fairness Levels ### Overview The image presents three causal diagrams, labeled "Level-One," "Level-Two," and "Level-Three," illustrating different levels of fairness considerations in a predictive model. Each diagram depicts relationships between protected attributes (e.g., race, sex), unfair observables (e.g., GPA, LSAT), a fair observable (Xfair), an outcome (FYA), and fair unobservables (epsilon GPA, epsilon LSAT, K). The diagrams use nodes to represent variables and arrows to indicate causal relationships. Dashed lines represent additive noise, and dashed circles indicate variables seen by the CFP (Counterfactual Fairness Prediction). ### Components/Axes * **Nodes:** Represent variables. * Blue: Protected Attributes (Prot. Attr) - SEX, RACE * Orange: Outcome - FYA (First Year Average) * Purple: Unfair Observable - GPA (Grade Point Average), LSAT (Law School Admission Test) * Yellow: Fair Observable - Xfair * Green: Fair Unobservable - K, epsilon GPA, epsilon LSAT * **Edges:** Represent relationships between variables. * Solid Arrow: Cause - Indicates a causal relationship. * Dashed Line: Additive Noise - Represents random variation or error. * **Circles with Dashed Outline:** Seen by CFP (Counterfactual Fairness Prediction). * **Titles:** * 1) Level-One * 2) Level-Two * 3) Level-Three * **Legend:** Located at the bottom of the image, explaining the node colors and edge types. ### Detailed Analysis or Content Details **1) Level-One:** * Nodes: GPA, LSAT, SEX, RACE, FYA, Xfair * Relationships: * SEX and RACE both have causal effects on GPA, LSAT, and FYA. * GPA and LSAT both have causal effects on FYA. * Xfair has a causal effect on FYA. * Xfair is seen by CFP. * FYA is seen by CFP. **2) Level-Two:** * Nodes: GPA, LSAT, SEX, RACE, FYA, K * Relationships: * SEX and RACE both have causal effects on GPA, LSAT, and FYA. * GPA and LSAT both have causal effects on FYA. * K has a causal effect on FYA. * K is seen by CFP. * FYA is seen by CFP. **3) Level-Three:** * Nodes: GPA, LSAT, SEX, RACE, FYA, epsilon GPA, epsilon LSAT * Relationships: * SEX and RACE both have causal effects on GPA, LSAT, and FYA. * GPA and LSAT both have causal effects on FYA. * epsilon GPA is additive noise to GPA. * epsilon LSAT is additive noise to LSAT. * FYA is seen by CFP. * epsilon GPA is seen by CFP. * epsilon LSAT is seen by CFP. ### Key Observations * The diagrams illustrate a progression in fairness considerations. * Level-One introduces a fair observable, Xfair, that directly influences the outcome. * Level-Two introduces a fair unobservable, K, that directly influences the outcome. * Level-Three introduces additive noise terms, epsilon GPA and epsilon LSAT, to GPA and LSAT respectively. * Protected attributes (SEX, RACE) consistently influence GPA, LSAT, and FYA across all levels. * The outcome, FYA, is always seen by the CFP. ### Interpretation The diagrams represent different approaches to achieving fairness in a predictive model. Level-One focuses on incorporating a fair observable feature, while Level-Two introduces a fair unobservable variable. Level-Three acknowledges the presence of noise or error in the unfair observable variables. The progression from Level-One to Level-Three suggests an increasing complexity in modeling and addressing fairness concerns. The consistent influence of protected attributes highlights the need to mitigate bias introduced by these variables. The fact that FYA is always seen by the CFP indicates that the model is always aware of the outcome, allowing for potential adjustments to improve fairness.

\n ## Diagram: Causal Fairness Levels ### Overview The image presents a diagram illustrating three levels of causal fairness, labeled Level-One, Level-Two, and Level-Three. Each level depicts a causal model with nodes representing variables and arrows indicating causal relationships. The diagram uses color-coding to distinguish between protected attributes, outcomes, observable and unobservable variables, and noise. The diagram aims to demonstrate how fairness can be achieved through different levels of intervention in a causal model. ### Components/Axes The diagram consists of three panels, arranged horizontally. Each panel represents a different level of fairness. The key components are: * **Nodes:** Represent variables such as SEX, RACE, GPA, LSAT, FYA, Xfair, and K. Nodes are color-coded as follows: * Blue: Protected Attribute (Prot. Attr.) * Orange: Outcome * Purple: Unfair Observable * Black: Unfair Unobservable * Green: Fair Unobservable * White: Seen by CFP (Counterfactual Fairness Policy) * **Arrows:** Represent causal relationships (Cause). Solid arrows indicate direct causal effects. * **Dotted Arrows:** Represent additive noise. * **Legend:** Located at the bottom of the image, explaining the color-coding and symbols used in the diagram. * **Level Labels:** "1) Level-One", "2) Level-Two", "3) Level-Three" are positioned above each panel. ### Detailed Analysis or Content Details **Level-One:** * Variables: SEX (blue), RACE (blue), GPA (purple), LSAT (purple), FYA (orange). * Causal relationships: * SEX -> GPA * SEX -> LSAT * RACE -> GPA * RACE -> LSAT * GPA -> FYA * LSAT -> FYA * Additive Noise: Dotted lines connect to GPA, LSAT, and FYA. **Level-Two:** * Variables: SEX (blue), RACE (blue), GPA (purple), LSAT (purple), FYA (orange), Xfair (orange). * Causal relationships: * SEX -> GPA * SEX -> LSAT * RACE -> GPA * RACE -> LSAT * GPA -> FYA * LSAT -> FYA * Xfair -> FYA * Unfair Unobservable: K (black) influences LSAT. * Additive Noise: Dotted lines connect to FYA. **Level-Three:** * Variables: SEX (blue), RACE (blue), GPA (purple), LSAT (purple), FYA (orange), εGPA (green), εLSAT (green). * Causal relationships: * SEX -> GPA * SEX -> LSAT * RACE -> GPA * RACE -> LSAT * GPA -> FYA * LSAT -> FYA * Fair Unobservable: εGPA influences GPA, εLSAT influences LSAT. * Additive Noise: Dotted lines connect to GPA, LSAT, and FYA. ### Key Observations * The complexity of the causal model increases from Level-One to Level-Three. * Level-Two introduces an unobservable variable (K) that influences LSAT, representing a source of unfairness. * Level-Three introduces fair unobservable variables (εGPA, εLSAT) that influence GPA and LSAT, respectively, suggesting a mechanism for achieving fairness. * The outcome variable (FYA) is directly influenced by multiple variables in each level. * The diagram illustrates a progression from a model with direct causal effects of protected attributes to a model where fairness is achieved through interventions on unobservable variables. ### Interpretation The diagram demonstrates a hierarchical approach to achieving causal fairness. Level-One represents a naive model where protected attributes (SEX, RACE) directly influence the outcome (FYA) through observable variables (GPA, LSAT). Level-Two introduces the concept of unobserved confounding (K), highlighting a source of unfairness that is not captured by simply controlling for observable variables. Level-Three proposes a solution by introducing fair unobservable variables (εGPA, εLSAT) that can be intervened upon to mitigate the effects of unfairness. The diagram suggests that achieving fairness requires a deeper understanding of the underlying causal mechanisms and the ability to intervene on unobservable variables. The CFP (Counterfactual Fairness Policy) is indicated as being able to "see" the white nodes, implying that it can leverage information about these variables to make fairer decisions. The progression from Level-One to Level-Three represents a shift from simply observing correlations to understanding and addressing the root causes of unfairness. The use of color-coding and symbols effectively communicates the different types of variables and their roles in the causal model. The diagram is a conceptual illustration of causal fairness and does not provide specific numerical data or quantitative analysis.

## Causal Diagram: Three-Level Fairness Model ### Overview The image displays three separate causal diagrams, labeled "1) Level-One", "2) Level-Two", and "3) Level-Three", arranged horizontally from left to right. These diagrams model the relationships between protected attributes, academic/test scores, and an outcome variable (FYA) under different fairness interventions. A comprehensive legend is provided at the bottom of the image to explain the node types and connection styles. ### Components/Axes (Legend) The legend at the bottom defines the following visual elements: * **Node Types (by color and border):** * **Blue circle (solid border):** "Prot. Attr" (Protected Attribute). Nodes: `SEX`, `RACE`. * **Orange circle (dashed border):** "Outcome". Node: `FYA`. * **Purple circle (solid border):** "Unfair Observable". Nodes: `GPA`, `LSAT`. * **Green circle (solid border):** "Fair Unobservable". Node: `K` (in Level-Two). * **Green circle (dashed border):** "Fair Unobservable". Nodes: `ε_GPA`, `ε_LSAT` (in Level-Three). * **Yellow circle (dashed border):** "Fair Observable". Node: `X_fair` (in Level-One). * **Connection Types:** * **Solid arrow (→):** "Cause". Indicates a direct causal influence. * **Dotted line (........):** "Additive Noise". Indicates the addition of a noise term. * **Dashed circle border:** "Seen by CFP". Indicates the variable is observed by the "CFP" (presumably a fairness-aware predictor or mechanism). ### Detailed Analysis #### **1) Level-One Diagram** * **Components:** `SEX`, `RACE` (Protected Attributes); `GPA`, `LSAT` (Unfair Observables); `FYA` (Outcome); `X_fair` (Fair Observable). * **Flow & Relationships:** * `SEX` and `RACE` have direct causal arrows pointing to `GPA`, `LSAT`, and `FYA`. * `GPA` has a causal arrow pointing to `LSAT`. * `GPA`, `LSAT`, and `X_fair` all have causal arrows pointing to `FYA`. * The `X_fair` node has a dashed border, indicating it is "Seen by CFP". * **Interpretation:** This level introduces a single, fair observable variable (`X_fair`) that directly influences the outcome (`FYA`). The protected attributes (`SEX`, `RACE`) still influence both the unfair observables (`GPA`, `LSAT`) and the outcome directly. #### **2) Level-Two Diagram** * **Components:** `SEX`, `RACE` (Protected Attributes); `GPA`, `LSAT` (Unfair Observables); `FYA` (Outcome); `K` (Fair Unobservable). * **Flow & Relationships:** * `SEX` and `RACE` have direct causal arrows pointing to `GPA`, `LSAT`, and `FYA`. * `GPA` has a causal arrow pointing to `LSAT`. * The new node `K` (Fair Unobservable, solid green circle) has causal arrows pointing to `GPA`, `LSAT`, and `FYA`. * The `K` node has a dashed border, indicating it is "Seen by CFP". * **Interpretation:** This level replaces the fair observable (`X_fair`) with a fair unobservable (`K`). This `K` variable influences all downstream variables (`GPA`, `LSAT`, `FYA`). The protected attributes still have their direct paths. #### **3) Level-Three Diagram** * **Components:** `SEX`, `RACE` (Protected Attributes); `GPA`, `LSAT` (Unfair Observables); `FYA` (Outcome); `ε_GPA`, `ε_LSAT` (Fair Unobservables/Noise). * **Flow & Relationships:** * `SEX` and `RACE` have direct causal arrows pointing to `GPA`, `LSAT`, and `FYA`. * `GPA` has a causal arrow pointing to `LSAT`. * Two new nodes, `ε_GPA` and `ε_LSAT` (Fair Unobservable, dashed green circles), are introduced. * `ε_GPA` is connected to `GPA` via a dotted line ("Additive Noise"). * `ε_LSAT` is connected to `LSAT` via a dotted line ("Additive Noise"). * Both `ε_GPA` and `ε_LSAT` have dashed borders, indicating they are "Seen by CFP". * **Interpretation:** This level models fairness by adding independent, fair noise terms (`ε`) to the unfair observable variables (`GPA`, `LSAT`). These noise terms are observed by the CFP. The direct influence of protected attributes on the observables and outcome remains. ### Key Observations 1. **Progression of Intervention:** The diagrams show a conceptual progression in fairness intervention: from adding a separate fair feature (Level-One), to introducing a single latent fair factor (Level-Two), to adding fair noise directly to the unfair observables (Level-Three). 2. **Persistent Bias Path:** In all three levels, the protected attributes (`SEX`, `RACE`) maintain direct causal links to both the intermediate variables (`GPA`, `LSAT`) and the final outcome (`FYA`). The interventions do not sever these paths but attempt to add "fair" components alongside them. 3. **"Seen by CFP" Condition:** The fairness mechanism (whether `X_fair`, `K`, or `ε` terms) is always marked as observable to the CFP, suggesting the fairness-aware model has access to these specific fair components. 4. **Node Styling Consistency:** The styling is consistent with the legend. Outcome (`FYA`) always has a dashed orange border. Protected attributes are always solid blue. Unfair observables are solid purple. The "fair" components vary in color (yellow/green) and border style (solid/dashed) based on their type and level. ### Interpretation These diagrams visually formalize different approaches to achieving fairness in a predictive model (e.g., for law school admission, given GPA and LSAT scores). The core problem is that protected attributes like `SEX` and `RACE` causally influence both the inputs (`GPA`, `LSAT`) and the outcome (`FYA`), leading to potential unfairness. * **Level-One** suggests fairness can be achieved by discovering and using an alternative, fair observable (`X_fair`) that captures legitimate predictive power without bias. * **Level-Two** posits the existence of a single, unobservable but fair latent factor (`K`) that drives both academic performance and the outcome. If this factor can be estimated or proxied, fairness might be attainable. * **Level-Three** takes a more statistical approach, suggesting that adding carefully calibrated, fair noise to the biased observables can "wash out" the unfair influence, making the final prediction fairer. The diagrams highlight a fundamental tension: the protected attributes have a real, causal effect on the variables we observe and the outcome. Fairness interventions, therefore, are not about erasing this reality but about strategically adding new, fair information or noise to counteract the *unfair* component of that influence during the decision-making process. The choice of model (Levels 1-3) implies different assumptions about what "fairness" means and what kind of additional data or noise is available.

## Causal Diagram: Three-Level Model of Fairness in Educational Outcomes ### Overview The diagram illustrates a three-level causal model examining how protected attributes (race, sex) influence educational outcomes (FYA) through observable/unobservable factors. It uses color-coded nodes and directional arrows to represent causal relationships, mediators, and error terms. ### Components/Axes - **Nodes**: - **Blue**: Protected Attributes (RACE, SEX) - **Purple**: Unfair Observables (GPA, LSAT) - **Orange**: Outcome (FYA) - **Yellow**: Fair Observable (X_fair) - **Green**: Fair Unobservable (K) - **Dashed Green**: Error Terms (ε_GPA, ε_LSAT) - **Arrows**: - Solid arrows: Direct causation - Dotted arrows: Additive noise - Dashed arrows: Causal paths seen by Conditional Fairness Preservation (CFP) - **Legend**: Located at the bottom, mapping colors to node types. ### Detailed Analysis #### Level-One - **Protected Attributes** (RACE, SEX) directly influence **Unfair Observables** (GPA, LSAT). - **Unfair Observables** collectively affect the **Outcome** (FYA) through a **Fair Observable** (X_fair). - **Additive Noise** (dotted lines) connects RACE/SEX to X_fair, indicating unmeasured confounding. #### Level-Two - Introduces **Fair Unobservable** (K) as a mediator between Unfair Observables (GPA, LSAT) and Outcome (FYA). - K is influenced by both GPA and LSAT, suggesting it captures latent fairness-related factors. - X_fair remains a direct predictor of FYA, now alongside K. #### Level-Three - Adds **Error Terms** (ε_GPA, ε_LSAT) as dashed green nodes, representing unobserved variability in GPA/LSAT. - These errors directly influence FYA, acknowledging measurement limitations or omitted variables. ### Key Observations 1. **Protected Attributes** (RACE, SEX) exert influence through both observable (GPA, LSAT) and unobservable pathways. 2. **Mediation by K**: The fair unobservable K partially mediates the effect of GPA/LSAT on FYA, suggesting latent fairness mechanisms. 3. **Error Terms**: Unobserved variability in GPA/LSAT (ε_GPA, ε_LSAT) directly impacts FYA, highlighting model uncertainty. 4. **Causal Flow**: Protected attributes → Unfair Observables → Outcome, with K and X_fair as fairness-correcting mediators. ### Interpretation This model demonstrates how protected attributes may indirectly affect educational outcomes through observable metrics (GPA, LSAT) and unobservable factors (K). The inclusion of K as a fairness unobservable suggests that traditional metrics alone may not capture equity dynamics. The error terms emphasize the limitations of relying solely on observed data. The diagram implies that fairness interventions should address both observable metrics (via X_fair) and latent factors (K) to mitigate bias in outcomes like FYA. The additive noise in Level-One indicates that racial/sexual disparities may persist even after accounting for observed variables, pointing to systemic biases beyond individual-level factors.