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Marginal Causal Flows for Validation and Inference

Daniel de Vassimon Manela, Laura Battaglia, Robin J. Evans

TL;DR

Frugal Flows address the challenge of validating causal inference methods by directly parameterising the marginal causal effect within a flexible, likelihood-based framework. By combining normalising flows for the data-generating process with copula-based conditioning, FFs learn the marginal outcome distribution under do(T) and the associated causal margin, while enabling exact specification of unobserved confounding and treatment heterogeneity. This yields realistic synthetic benchmarks that closely resemble real-world data yet encode user-defined causal properties, improving robustness checks for causal methods. The approach advances causal benchmarking by providing precise control over ATE, confounding, and overlap, with demonstrated benefits in simulated and real-data experiments, at the cost of higher data and tuning requirements. FFs thus offer a principled, configurable platform for validating and stress-testing causal inference methods in complex data settings.

Abstract

Investigating the marginal causal effect of an intervention on an outcome from complex data remains challenging due to the inflexibility of employed models and the lack of complexity in causal benchmark datasets, which often fail to reproduce intricate real-world data patterns. In this paper we introduce Frugal Flows, a novel likelihood-based machine learning model that uses normalising flows to flexibly learn the data-generating process, while also directly inferring the marginal causal quantities from observational data. We propose that these models are exceptionally well suited for generating synthetic data to validate causal methods. They can create synthetic datasets that closely resemble the empirical dataset, while automatically and exactly satisfying a user-defined average treatment effect. To our knowledge, Frugal Flows are the first generative model to both learn flexible data representations and also exactly parameterise quantities such as the average treatment effect and the degree of unobserved confounding. We demonstrate the above with experiments on both simulated and real-world datasets.

Marginal Causal Flows for Validation and Inference

TL;DR

Frugal Flows address the challenge of validating causal inference methods by directly parameterising the marginal causal effect within a flexible, likelihood-based framework. By combining normalising flows for the data-generating process with copula-based conditioning, FFs learn the marginal outcome distribution under do(T) and the associated causal margin, while enabling exact specification of unobserved confounding and treatment heterogeneity. This yields realistic synthetic benchmarks that closely resemble real-world data yet encode user-defined causal properties, improving robustness checks for causal methods. The approach advances causal benchmarking by providing precise control over ATE, confounding, and overlap, with demonstrated benefits in simulated and real-data experiments, at the cost of higher data and tuning requirements. FFs thus offer a principled, configurable platform for validating and stress-testing causal inference methods in complex data settings.

Abstract

Investigating the marginal causal effect of an intervention on an outcome from complex data remains challenging due to the inflexibility of employed models and the lack of complexity in causal benchmark datasets, which often fail to reproduce intricate real-world data patterns. In this paper we introduce Frugal Flows, a novel likelihood-based machine learning model that uses normalising flows to flexibly learn the data-generating process, while also directly inferring the marginal causal quantities from observational data. We propose that these models are exceptionally well suited for generating synthetic data to validate causal methods. They can create synthetic datasets that closely resemble the empirical dataset, while automatically and exactly satisfying a user-defined average treatment effect. To our knowledge, Frugal Flows are the first generative model to both learn flexible data representations and also exactly parameterise quantities such as the average treatment effect and the degree of unobserved confounding. We demonstrate the above with experiments on both simulated and real-world datasets.

Paper Structure

This paper contains 47 sections, 2 theorems, 28 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Marginal Causal Models
  4. Frugal Parameterisations
  5. Variation Independence
  6. Copulae
  7. Copulae in Machine Learning
  8. Paper Motivation
  9. Normalising Flows
  10. Copula Flows
  11. Validating and Benchmarking Causal Methods
  12. Method
  13. Building the Joint Distribution
  14. Constructing Frugal Flows
  15. Learning the Propensity Flow
  16. ...and 32 more sections

Key Result

Theorem 1

For a d-variate distribution function $F_{1:d} \in \mathcal{F}(F_1,\ldots,F_d)$, with $j^{\text{th}}$ univariate margin $F_j$, the copula associated with $F$ is a distribution function $C : [0,1]^d \rightarrow[0,1]$ with uniform margins on $(0,1)$ that satisfies The copula distribution is associated with its density $c(\cdot)$ where $f_i(\cdot)$ is the univariate density function of the $i^{\text

Figures (10)

  • Figure 1: A visual abstract outlining the different components of a frugal model, and how each specific component is parameterised. Univariate CDFs are denoted by $F$, and copula distribution functions are denoted by $C$. The marginal causal effect, $\theta_{{Y\space|\space \operatorname{do}(T)}}$, is modelled with a univariate normalising flow, which we denote by $\mathcal{F}$ (see \ref{['subsec:normalising-flows']}). The intervened dependency measure, $\phi_{{\bm{Z}\space Y \space | \space\operatorname{do}(T)}}$, is modelled with a copula flow which we denote by $\mathcal{C}$ (see \ref{['subsec:copula-flows']}). The past, $\theta_{{\space Z \space T}}$, is modelled by the combination of univariate normalising flows (for the univariate pretreatment covariate distributions) and a copula flow (for the propensity of treatment).
  • Figure 2: Structure for learning a Frugal Flow. The top line outlines the process for learning the univariate marginal flows of the pretreatment covariates $\bm{Z}$. The bottom transform illustrates the Frugal Flow, which learns the conditional copula $c(\bm{v}_{\bm{Z}}\mid v_{{Y\space|\space \operatorname{do}(T)}})$ jointly with the causal marginal flow $\mathcal{F}_{{Y\space|\space \operatorname{do}(T)}}$ by enforcing $V_{{Y\space|\space \operatorname{do}(T)}}$ to be marginally uniform.
  • Figure 3: Boxplot of ATE estimates from 10 inference methods, estimated across 50 different samples from a FF trained on the Lalonde dataset. The dotted red line represents the customized ATE of samples generated from the trained Frugal Flow.
  • Figure 4: Boxplot of ATE estimates from 10 inference methods, estimated across 50 different samples from a FF trained on the e401(k) dataset. The dotted red line represents the customized ATE of samples generated from the trained Frugal Flow.
  • Figure 5: A generalised example of a static causal treatment model. The past $P(T, \bm{Z})$ (black) can be freely specified separately from the causal effect (blue). However, the dependency measure between $\bm{Z}$ and $Y$ (red), $\phi$ should be parameterised in such a way that the margins $P(\bm{Z})$ and $P({Y\space|\space \operatorname{do}(T)})$ are invariant to changes in $\phi$.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2