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Sequential Conditional Transport on Probabilistic Graphs for Interpretable Counterfactual Fairness

Agathe Fernandes Machado, Arthur Charpentier, Ewen Gallic

TL;DR

The paper tackles counterfactual fairness by marrying causal graphs with optimal transport to generate interpretable counterfactuals. It develops sequential transport on probabilistic graphical models, extending Knothe-Rosenblatt maps to DAGs so counterfactuals can be computed via a sequence of univariate conditional transports $T_i^\ star$, i.e., $T_{\mathcal{G}}^*(x_1,...,x_d)=\prod_i T_i^*(x_i|\mathrm{parents}(x_i))$, with $T_i^* = F_{i|Pa}^{-1}\circ F_{i|Pa}$, guaranteeing tractable, closed-form updates. The approach yields an out-of-sample estimation procedure for individual counterfactuals and enables global fairness assessments through metrics like counterfactual demographic parity and related indices, demonstrated on synthetic data and real datasets where it aligns with or surpasses existing methods like FairAdapt. Overall, sequential transport provides an interpretable, graph-consistent framework for counterfactual fairness with practical impact for auditing and mitigating discrimination in predictive models.

Abstract

In this paper, we link two existing approaches to derive counterfactuals: adaptations based on a causal graph, and optimal transport. We extend "Knothe's rearrangement" and "triangular transport" to probabilistic graphical models, and use this counterfactual approach, referred to as sequential transport, to discuss fairness at the individual level. After establishing the theoretical foundations of the proposed method, we demonstrate its application through numerical experiments on both synthetic and real datasets.

Sequential Conditional Transport on Probabilistic Graphs for Interpretable Counterfactual Fairness

TL;DR

The paper tackles counterfactual fairness by marrying causal graphs with optimal transport to generate interpretable counterfactuals. It develops sequential transport on probabilistic graphical models, extending Knothe-Rosenblatt maps to DAGs so counterfactuals can be computed via a sequence of univariate conditional transports , i.e., , with , guaranteeing tractable, closed-form updates. The approach yields an out-of-sample estimation procedure for individual counterfactuals and enables global fairness assessments through metrics like counterfactual demographic parity and related indices, demonstrated on synthetic data and real datasets where it aligns with or surpasses existing methods like FairAdapt. Overall, sequential transport provides an interpretable, graph-consistent framework for counterfactual fairness with practical impact for auditing and mitigating discrimination in predictive models.

Abstract

In this paper, we link two existing approaches to derive counterfactuals: adaptations based on a causal graph, and optimal transport. We extend "Knothe's rearrangement" and "triangular transport" to probabilistic graphical models, and use this counterfactual approach, referred to as sequential transport, to discuss fairness at the individual level. After establishing the theoretical foundations of the proposed method, we demonstrate its application through numerical experiments on both synthetic and real datasets.
Paper Structure (33 sections, 1 theorem, 45 equations, 19 figures, 3 tables, 5 algorithms)

This paper contains 33 sections, 1 theorem, 45 equations, 19 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Graphical Models and Causal Networks
  4. Probabilistic Graphical Models
  5. Causal Networks and Linear Structural Models
  6. Intervention in a Linear Structural Model
  7. Non-Linear Structural Models
  8. Presentation of the Model
  9. Connections With Conditional Quantiles
  10. Optimal Transport
  11. Univariate Optimal Transport
  12. Multivariate Optimal Transport
  13. Sequential Transport
  14. Knothe-Rosenblatt Conditional Transport
  15. Sequential Conditional Transport on a Probabilistic Graph
  16. ...and 18 more sections

Key Result

Proposition A.1

If $\boldsymbol{X}\sim\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$, with notations of Eq. eq:gaussian, $\boldsymbol{B}=\boldsymbol{\Sigma}^{-1}$, $\boldsymbol{X}$ is Markov with respect to $\mathcal{G}=(E,V)$ if and only if $B_{i,j}=0$ whenever $(i,j),(j,i)\notin E$.

Figures (19)

  • Figure 1: Causal graph in the German Credit dataset from watson2021local, or DAG.
  • Figure 2: Linear Structural Causal Model -- observation.
  • Figure 3: Linear Structural Causal Model -- intervention.
  • Figure 4: Univariate OT, with Gaussian distributions (left) and general marginal distributions (right). The transport curve ($T^\star$) is shown in red.
  • Figure 5: Two simple causal networks, with two legitimate mitigating variables, $x_1$ and $x_2$.
  • ...and 14 more figures

Theorems & Definitions (4)

  • Definition
  • Remark
  • Proposition A.1
  • proof