## Causal Diagram: Six Fairness Models in Machine Learning ### Overview The image displays six distinct causal models, numbered 1 through 6, illustrating different relationships between a protected attribute (`A`), observable features (`X_b`, `X_f`), an unobservable feature (`U`), an outcome (`Y`), and additive noise terms (`ε`). Each model is presented as a panel containing a directed acyclic graph (DAG) and its corresponding mathematical formulation. A legend at the bottom defines the visual symbols and node colors. ### Components/Axes The image is organized into six horizontal panels. Each panel contains: 1. **Title**: A numbered label (e.g., "1) Biased"). 2. **Causal Graph**: A diagram with colored nodes and arrows. 3. **Mathematical Formulation**: A set of equations defining the distributions and relationships between variables. **Legend (Bottom of Image):** * **Node Colors & Meanings**: * Blue Circle: `Prot. Attr.` (Protected Attribute, `A`) * Orange Circle: `Outcome` (`Y`) * Purple Circle: `Unfair Observable` (`X_b`) * Yellow Circle: `Fair Observable` (`X_f`) * Green Circle: `Fair Unobservable` (`U`) * **Arrow Types**: * Solid Arrow (`→`): `Cause` * Dotted Arrow (`⋯>`): `Additive Noise` * **Node Border**: * Dashed Circle Border: `Non-descendent` * **Node Fill Pattern**: * Hatched Pattern: `Seen by FairPFN` ### Detailed Analysis #### Panel 1: Biased * **Graph**: `A` (blue, hatched) → `X_b` (purple, hatched) → `Y` (orange, hatched). `X_b` has an additive noise term `ε_Xb` (green). `Y` has an additive noise term `ε_Y` (orange). * **Equations**: * `A ~ U({0,1})` * `ε_Xb, ε_Y ~ N(μ, σ), N(μ, σ)` * `X_b = w_A * A² + ε_Xb` * `Y = w_Xb * X_b² + ε_Y` * `Y = 1(Y ≥ Ȳ)` #### Panel 2: Direct-Effect * **Graph**: `A` (blue, hatched) → `X_f` (yellow, hatched) → `Y` (orange, hatched). `A` also has a direct arrow to `Y`. `X_f` has an additive noise term `ε_Xf` (green). `Y` has an additive noise term `ε_Y` (orange). * **Equations**: * `A ~ U({0,1})` * `ε_Xb, ε_Y ~ N(μ, σ), N(μ, σ)` * `X_f = N(μ, σ)` * `Y = w_Xf * X_f² + w_A * A² + ε_Y` * `Y = 1(Y ≥ Ȳ)` #### Panel 3: Indirect-Effect * **Graph**: `A` (blue, hatched) → `X_b` (purple, hatched) → `Y` (orange, hatched). `X_f` (yellow, dashed border) → `Y`. `X_b` has an additive noise term `ε_Xb` (green). `Y` has an additive noise term `ε_Y` (orange). * **Equations**: * `ε_Xb, ε_Y ~ N(μ, σ), N(μ, σ)` * `A ~ U({0,1}), X_f ~ N(μ, σ)` * `X_b = w_A * A² + ε_Xb` * `Y = w_Xb * X_b² + w_Xf * X_f² + ε_Y` * `Y = 1(Y ≥ Ȳ)` #### Panel 4: Fair Observable * **Graph**: `A` (blue, hatched) → `X_b` (purple, hatched) → `Y` (orange, hatched). `A` also has a direct arrow to `Y`. `X_f` (yellow, dashed border) → `Y`. `X_b` has an additive noise term `ε_Xb` (green). `Y` has an additive noise term `ε_Y` (orange). * **Equations**: * `ε_Xb, ε_Y ~ N(μ, σ), N(μ, σ)` * `A ~ U({0,1}), X_f ~ N(μ, σ)` * `X_b = w_A * A² + ε_Xb` * `Y = w_Xb * X_b² + w_Xf * X_f² + w_A * A² + ε_Y` * `Y = 1(Y ≥ Ȳ)` #### Panel 5: Fair Unobservable * **Graph**: `A` (blue, hatched) → `X_b` (purple, hatched) → `Y` (orange, hatched). `A` also has a direct arrow to `Y`. `U` (green, dashed border) → `X_b` and `U` → `Y`. `X_b` has an additive noise term `ε_Xb` (green). `Y` has an additive noise term `ε_Y` (orange). * **Equations**: * `ε_Xb, ε_Y ~ N(μ, σ), N(μ, σ)` * `A ~ U({0,1}), U ~ N(μ, σ)` * `X_b = w_A * A² + w_U * U² + ε_Xb` * `Y = w_Xb * X_b² + w_A * A² + ε_Y` * `Y = 1(Y ≥ Ȳ)` #### Panel 6: Fair Additive Noise * **Graph**: `A` (blue, hatched) → `X_b` (purple, hatched) → `Y` (orange, hatched). `A` also has a direct arrow to `Y`. `X_b` has an additive noise term `ε_Xb` (green). `Y` has an additive noise term `ε_Y` (orange). * **Equations**: * `ε_Xb, ε_Y ~ N(μ, σ), N(μ, σ)` * `A ~ U({0,1})` * `X_b = w_A * A² + ε_Xb` * `Y = w_Xb * X_b² + w_A * A² + ε_Y` * `Y = 1(Y ≥ Ȳ)` ### Key Observations 1. **Progression of Complexity**: The models progress from a simple biased pathway (1) to more complex structures incorporating direct effects (2, 4, 6), indirect effects (3), and unobservable confounders (5). 2. **Variable Roles**: The protected attribute `A` is always binary (`U({0,1})`). The outcome `Y` is always binarized via a threshold (`1(Y ≥ Ȳ)`). Observable features (`X_b`, `X_f`) and the unobservable `U` are modeled with normal distributions. 3. **Visual Coding**: The hatched pattern indicates which variables are "Seen by FairPFN," suggesting this diagram is from a paper proposing or analyzing a method called FairPFN. The dashed border for `X_f` and `U` in panels 3, 4, and 5 marks them as "Non-descendent" of `A` in those specific causal structures. 4. **Mathematical Consistency**: All models use squared terms (e.g., `A²`, `X_b²`) in their structural equations, implying non-linear relationships. The noise terms are consistently modeled as Gaussian. ### Interpretation This diagram is a technical taxonomy of data-generating processes used to study algorithmic fairness. It systematically varies the causal pathways through which a protected attribute (`A`) can influence an outcome (`Y`). * **Model 1 (Biased)** represents a scenario where bias flows entirely through a single, unfair observable feature (`X_b`). * **Models 2 & 6 (Direct-Effect, Fair Additive Noise)** introduce a direct effect of `A` on `Y`, which is often considered a source of unfair discrimination. * **Model 3 (Indirect-Effect)** separates features into fair (`X_f`) and unfair (`X_b`) observables, with `A` only affecting `Y` through the unfair one. * **Model 4 (Fair Observable)** combines direct effect with both fair and unfair observable pathways. * **Model 5 (Fair Unobservable)** is the most complex, introducing an unobserved confounder (`U`) that affects both the unfair feature and the outcome, representing real-world complexity where important factors are not measured. The core purpose is to provide a framework for evaluating fairness interventions (like the mentioned "FairPFN"). By defining these precise data-generating models, researchers can test whether a fairness algorithm works correctly under different, well-specified assumptions about how bias enters a system. The consistent use of squared terms and Gaussian noise creates a controlled, synthetic environment for this evaluation. The diagram argues that understanding fairness requires moving beyond simple correlations to explicitly model the causal structure of the data.