Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sequential Transport for Causal Mediation Analysis

Agathe Fernandes-Machado, Iryna Voitsitska, Arthur Charpentier, Ewen Gallic

Abstract

We propose sequential transport (ST), a distributional framework for mediation analysis that combines optimal transport (OT) with a mediator directed acyclic graph (DAG). Instead of relying on cross-world counterfactual assumptions, ST constructs unit-level mediator counterfactuals by minimally transporting each mediator, either marginally or conditionally, toward its distribution under an alternative treatment while preserving the causal dependencies encoded by the DAG. For numerical mediators, ST uses monotone (conditional) OT maps based on conditional CDF/quantile estimators; for categorical mediators, it extends naturally via simplex-based transport. We establish consistency of the estimated transport maps and of the induced unit-level decompositions into mutatis mutandis direct and indirect effects under standard regularity and support conditions. When the treatment is randomized or ignorable (possibly conditional on covariates), these decompositions admit a causal interpretation; otherwise, they provide a principled distributional attribution of differences between groups aligned with the mediator structure. Gaussian examples show that ST recovers classical mediation formulas, while additional simulations confirm good performance in nonlinear and mixed-type settings. An application to the COMPAS dataset illustrates how ST yields deterministic, DAG-consistent counterfactual mediators and a fine-grained mediator-level attribution of disparities.

Sequential Transport for Causal Mediation Analysis

Abstract

We propose sequential transport (ST), a distributional framework for mediation analysis that combines optimal transport (OT) with a mediator directed acyclic graph (DAG). Instead of relying on cross-world counterfactual assumptions, ST constructs unit-level mediator counterfactuals by minimally transporting each mediator, either marginally or conditionally, toward its distribution under an alternative treatment while preserving the causal dependencies encoded by the DAG. For numerical mediators, ST uses monotone (conditional) OT maps based on conditional CDF/quantile estimators; for categorical mediators, it extends naturally via simplex-based transport. We establish consistency of the estimated transport maps and of the induced unit-level decompositions into mutatis mutandis direct and indirect effects under standard regularity and support conditions. When the treatment is randomized or ignorable (possibly conditional on covariates), these decompositions admit a causal interpretation; otherwise, they provide a principled distributional attribution of differences between groups aligned with the mediator structure. Gaussian examples show that ST recovers classical mediation formulas, while additional simulations confirm good performance in nonlinear and mixed-type settings. An application to the COMPAS dataset illustrates how ST yields deterministic, DAG-consistent counterfactual mediators and a fine-grained mediator-level attribution of disparities.
Paper Structure (62 sections, 10 theorems, 137 equations, 10 figures, 2 tables, 4 algorithms)

This paper contains 62 sections, 10 theorems, 137 equations, 10 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Causal mediation analysis
  3. Potential outcomes and effect decomposition
  4. Motivation for distributional counterfactuals
  5. Optimal transport theory for a mutatis mutandis counterfactual mechanism
  6. Transport-based counterfactuals in the literature
  7. Mutatis mutandis semantics
  8. Transport maps and individual-level counterfactuals
  9. Transport-induced mutatis mutandis estimands
  10. Strengths and limitations of transport-based approaches
  11. Sequential transport on a mediator DAG
  12. Mediator DAG and notation
  13. Sequential transport
  14. Transport for numerical mediators
  15. Transport for categorical mediators
  16. ...and 47 more sections

Key Result

Lemma 5.1

Let $j$ be such that $\mathrm{parents}_{\mathcal{G}}(X_j)=\emptyset$ (so $\boldsymbol Z_j$ is empty). Under Assumptions A1--A3, the estimated marginal transport map is consistent: in the numerical case, for any compact $\mathcal{X}_j$ contained in the interior of $\mathrm{supp}(X_j\mid A=0)$.

Figures (10)

  • Figure 1: DAG for the simulated data.
  • Figure 2: Estimation (200 replicated samples, $n=600$) of average indirect effect $\bar{\delta}$, average direct effect $\bar{\zeta}$ and average total causal effect $\bar{\tau}$.
  • Figure 3: Simplified DAG for the COMPAS dataset.
  • Figure 4: Individual mutatis mutandis effects (distributional decomposition) for the COMPAS dataset estimated with counterfactual-based methods using Fairadapt (FPT-RF), optimal transport (OT), penalized OT (SKH), and sequential transport (ST) according to the causal graph of Figure \ref{['fig:dag-compas-example']}.
  • Figure 5: Untreated units (green dots), treated Units (yellow dots), and transported units (blue triangles) under optimal transport (OT), entropic regularization with Sinkhorn algorithm (SKH) and sequential transport: ST(1) (transporting $X_1$ first and then $X_1\to X_2$) and ST(2) (transporting $X_2$ first and then $X_2\to X_1$). Lines indicate the transport mappings.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Lemma 5.1: Consistency for numeric root mediators
  • Proposition 5.1: Consistency of conditional numerical transport
  • Corollary 5.1: Sequential (parent-transported) version
  • Proposition 5.2: Plug-in along estimated transported parents
  • Proposition 5.3: Pointwise consistency of simplex transport for categorical mediators
  • Corollary 5.2: Consistency along the sequential transport path
  • Proposition 5.4: Plug-in consistency of transport along estimated sequential transport
  • Theorem 5.1: Consistency of sequential mediator counterfactuals (ST)
  • Lemma 5.2: Consistency of induced categorical assignments
  • Remark : Dependence on the topological order
  • ...and 11 more