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Sequential Transport for Causal Mediation Analysis

This paper proposes Sequential Transport (ST), a distributional framework for causal mediation analysis that constructs unit-level mediator counterfactuals by minimally transporting mediators toward alternative treatment distributions while preserving causal dependencies via a DAG, thereby enabling consistent estimation of direct and indirect effects without relying on cross-world counterfactual assumptions.

Original authors: Agathe Fernandes-Machado, Iryna Voitsitska, Arthur Charpentier, Ewen Gallic

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Agathe Fernandes-Machado, Iryna Voitsitska, Arthur Charpentier, Ewen Gallic

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery: Why do two groups of people end up with different outcomes?

Let's say Group A (let's call them "Team Blue") and Group B ("Team Red") are applying for jobs. Team Blue gets hired much more often than Team Red. You want to know: Is this because of discrimination (the direct effect), or is it because Team Red has less experience or fewer qualifications (the indirect effect)?

This is the classic problem of Causal Mediation Analysis. For a long time, statisticians tried to solve this by building complex mathematical models (like a rigid blueprint) to guess what would have happened if a person from Team Red had Team Blue's qualifications. But these blueprints often fail because the real world is messy, non-linear, and full of surprises.

This paper introduces a new, smarter detective tool called Sequential Transport (ST). Here is how it works, explained through a simple story.

The Problem with "Time Travel"

Traditional methods rely on "cross-world counterfactuals." Imagine you have to time-travel a specific person from Team Red to a parallel universe where they were born into Team Blue. In that universe, you have to guess exactly how their life would have changed.

The Flaw: You can't actually time travel. To make the guess, you have to assume a specific "law of physics" (a structural model) for how the world works. If your law of physics is wrong, your guess is wrong.

The New Solution: "The Minimal Edit"

The authors propose a different approach called Sequential Transport. Instead of time travel, think of it as a smart, minimal edit to a person's resume.

Imagine you have a resume for a person from Team Red. You want to see what their resume would look like if they were in Team Red's distribution (the general pool of Team Red people) but had the opportunities of Team Blue.

The Golden Rule: Change the resume as little as possible to make it look like it belongs to Team Blue, but only change the parts that Team Blue actually influences.

The "Assembly Line" Analogy (The DAG)

The paper introduces a crucial concept: the DAG (Directed Acyclic Graph). Think of this as a flowchart or an assembly line of cause and effect.

Maybe being Black (Treatment) affects your Age at hiring.
Your Age affects your Number of Prior Jobs.
Your Prior Jobs affect your Final Score.

In the old methods, if you tried to change the "Final Score" to match Team Blue, you might accidentally break the logic of the assembly line. You might make someone 50 years old with 0 jobs, which is impossible in reality.

Sequential Transport fixes this by moving down the assembly line, one step at a time:

Step 1: Look at the first item on the list (e.g., Age). Gently nudge the person's age to match the average age of Team Blue.
Step 2: Now, look at the next item (Prior Jobs). But wait! You must calculate the "nudge" for Prior Jobs based on the new Age you just set. You don't just guess; you follow the rules of the assembly line.
Step 3: Move to the next item, using the results from the previous steps.

By doing this sequentially, the method ensures that the "counterfactual" person (the edited resume) is still a realistic human being who fits the rules of the world, even though they have been tweaked to look like they belong to the other group.

Handling Different "Types" of Data

The paper also solves a tricky problem: What if some data is numbers (like salary) and some is categories (like "High School," "College," "PhD")?

For Numbers: They use a smooth sliding scale to shift values up or down.
For Categories: They use a "probability map." Instead of just flipping a coin to change "High School" to "PhD," they calculate the exact probability of that change happening and then make a deterministic choice that keeps the overall statistics fair.

Why Does This Matter?

No Rigid Blueprints: You don't need to assume the world follows a straight line (linear). The method learns the shape of the data directly.
Individual Level: It doesn't just tell you the average difference. It tells you exactly why this specific person got a lower score. Was it because of their age? Their prior jobs? Or was it something else?
Fairness: In the real-world test (using the COMPAS dataset, which predicts criminal recidivism), the method showed that about two-thirds of the racial disparity in scores came from differences in "mediators" (like prior criminal records), while the rest was direct. This gives policymakers a clear target: "If we fix the distribution of prior records, we fix most of the unfairness."

The Bottom Line

Think of Sequential Transport as a smart editor for reality.

Old Way: "If you were Team Blue, you would be this totally different person." (Requires guessing the laws of the universe).
New Way (ST): "If you were Team Blue, here is the minimal amount we would need to change about your background to fit in, while keeping your life story logical and consistent."

It allows us to decompose complex inequalities into clear, understandable parts, helping us see exactly where the unfairness lies and how to fix it, without needing to build a perfect model of the entire universe.

1. Problem Statement

Causal mediation analysis aims to decompose the total effect of a treatment ( $A$ ) on an outcome ( $Y$ ) into direct effects and indirect effects transmitted through mediators ( $X$ ).

Limitations of Classical Approaches: Traditional methods rely on Structural Causal Models (SCMs), often assuming linear Structural Equation Models (SEMs) or specific parametric forms. These approaches struggle with:
- High-dimensional or mixed-type mediators: Handling continuous, categorical, and ordinal variables simultaneously.
- Unit-level decomposition: Most methods provide population-level averages rather than individual-level counterfactuals.
- Cross-world counterfactuals: Defining individual effects (e.g., $Y(0, X(1))$ ) requires untestable assumptions about unobserved structural noise and functional forms, which are often not identifiable from observational data.
The Gap: There is a need for a non-parametric, distributional framework that constructs unit-level mediator counterfactuals without specifying a full SCM, while respecting the causal dependencies encoded in a Directed Acyclic Graph (DAG).

2. Methodology: Sequential Transport (ST)

The authors propose Sequential Transport (ST), a framework that combines Optimal Transport (OT) with the structure of a mediator DAG to construct "mutatis mutandis" counterfactuals.

Core Concept: Mutatis Mutandis

Instead of assuming a structural equation $X(1) = f(1, U)$ , ST asks: How should an individual's mediator vector change minimally to align with the distribution of mediators under a different treatment, while respecting the causal dependencies between mediators?

The Algorithm

ST constructs counterfactual mediators by transporting them sequentially along a topological ordering of the mediator DAG ( $G$ ).

Topological Ordering: The mediators $X_1, \dots, X_d$ are ordered such that parents of any node appear before the node itself.
Sequential Mapping: For an untreated unit ( $A=0$ $A = 0$ ) with factual mediators $x_0$ $x_{0}$ , the counterfactual $x^\star_1$ $x_{1}^{⋆}$ is constructed recursively:
- Root Nodes (No parents): Apply a univariate OT map to transform the marginal distribution of $X_j|A=0$ to $X_j|A=1$ .
- Child Nodes: Apply a conditional OT map. The transport of $X_j$ depends on its parents. Crucially, the parents are first transported to their counterfactual values ( $x^\star_{1, \text{parents}}$ ). The map transforms $X_j$ conditional on the transported parent values to match the target conditional distribution.
- Formula: $x^\star_{1, \pi(j)} = T_{\pi(j)}(x_{0, \pi(j)} \mid x_{0, \text{parents}}, x^\star_{1, \text{parents}})$ .

Handling Data Types

Numerical Mediators: Uses monotone transport maps based on conditional Cumulative Distribution Functions (CDFs) and Quantile functions.
- $T(x) = Q_1(F_0(x))$ , where $F_0$ is the source CDF and $Q_1$ is the target quantile function.
- For conditional cases, kernel-based estimators approximate the conditional CDF and quantile functions.
Categorical Mediators: Treats categorical variables as probability vectors on a simplex.
- Estimates conditional probability vectors (e.g., via multinomial regression).
- Applies optimal transport on the simplex to map source probabilities to target probabilities.
- Converts the transported probability vector back to a discrete label (e.g., via argmax or semi-discrete OT allocation) to preserve marginal distributions.

Effect Decomposition

Once the counterfactual mediator vector $x^\star_1$ is constructed for an untreated unit, the effects are defined using outcome regression models $\hat{\mu}_a(x) = E[Y|A=a, X=x]$ :

Indirect Effect ( $\delta_i$ ): $\hat{\mu}_0(x^\star_1) - \hat{\mu}_0(x_0)$ (Change due to mediator shift).
Direct Effect ( $\zeta_i$ ): $\hat{\mu}_1(x^\star_1) - \hat{\mu}_0(x^\star_1)$ (Change due to treatment, holding mediator fixed at counterfactual).
Total Effect ( $\tau_i$ ): Sum of the two.

3. Key Contributions

Unit-Level DAG-Aligned Counterfactuals: ST provides a deterministic, interpretable method to generate individual-level mediator counterfactuals that are consistent with the causal DAG structure, avoiding the "cross-world" identification issues of traditional SEMs.
Theoretical Guarantees: The authors establish consistency results for the estimated transport maps and the induced effect decompositions. They prove that as sample size increases, the estimated sequential transport converges to the population-level mutatis mutandis transformation under standard regularity conditions (kernel convergence, smoothness of conditional distributions).
Mixed-Type Flexibility: The framework naturally handles mixed data types (continuous, categorical, ordinal) within a single DAG structure, a significant improvement over linear SEMs.
Distributional vs. Causal Interpretation: The paper clearly distinguishes between the distributional definition of ST (always valid if distributions are estimable) and its causal interpretation (valid only under sequential ignorability/randomization).

4. Results

Gaussian Simulations: In linear Gaussian settings, ST perfectly recovers the classical mediation formulas (e.g., $\alpha\beta$ for indirect effects), validating its consistency with standard theory.
Monte Carlo Simulations: In non-linear, mixed-type settings (continuous and categorical mediators, binary outcomes), ST demonstrates:
- Stability: Lower dispersion in effect estimates compared to global multivariate OT.
- Robustness: Performs well even when the data generating process is non-linear, outperforming linear SEMs in these regimes.
- Accuracy: Average effects align closely with benchmarks, while providing unique unit-level decompositions.
Real-World Application (COMPAS Dataset):
- Context: Analyzing racial disparities in recidivism predictions.
- Findings: ST attributes approximately two-thirds of the disparity to differences in mediator distributions (indirect effect) and one-third to direct effects.
- Granularity: It identifies specific mediators (e.g., prior convictions, charge degree) driving the indirect effect, offering a fine-grained attribution of disparities that global methods miss.

5. Significance and Implications

Bridging Causal Inference and Optimal Transport: The paper successfully integrates OT, traditionally used for distribution alignment, into the rigorous framework of causal mediation analysis.
Algorithmic Fairness: ST offers a powerful tool for fairness auditing. It allows practitioners to decompose disparities into specific causal pathways (mediators) without assuming a specific functional form for the data, making it robust to model misspecification.
Interpretability: By providing deterministic counterfactuals for each individual, ST enables "what-if" analysis at the individual level (e.g., "If this specific defendant had the prior record distribution of the white group, how would their risk score change?").
Practical Utility: The method is applicable to complex, real-world datasets with heterogeneous variable types, moving beyond the restrictive assumptions of linear structural models.

In summary, Sequential Transport offers a principled, non-parametric, and graph-consistent alternative to traditional mediation analysis, providing robust unit-level insights into how treatment effects are transmitted through intermediate variables.