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Marginal Causal Flows for Validation and Inference
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Introduces Frugal Flows, a normalising-flow copula model that jointly learns covariate copulas and the causal margin so that user-specified ATEs and unobserved confounding can be exactly imposed when generating synthetic causal benchmarks.
Problem
Generating realistic synthetic causal benchmark datasets in which the marginal causal effect (ATE), confounding, and overlap can be exactly specified is hard, since existing generators enforce causal constraints only via soft optimisation and need post hoc checks.
Approach
Combine the frugal parameterisation with normalising flows: a copula flow over covariate ranks and an outcome-margin flow ordered first so that the causal margin is learned jointly with the copula, making the ATE and confounding exact by construction.
Key results
- On simulated models M1-M3 with ATE 1 and 5, Frugal Flow estimates lie within the 2-sigma interval of the true ATE in most settings (e.g., M2 ATE 1: 1.01 ± 0.10), while outcome regression is biased (e.g., 1.46 ± 0.07).
- Frugal Flows generate samples whose ATE exactly matches the user-specified value by construction (e.g., ATE 1000 on Lalonde and 401(k) benchmarks), without post hoc checks.
- Logistic-outcome benchmark: IPW recovers the true logistic parameters (-0.88 ± 0.38 vs -1 truth; 1.6 ± 0.48 vs 2) whereas outcome regression is biased (-1.59, 3.16).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Frugal Flow instead of outcome regression | Causal ATE inference · M1, true ATE 1 | ATE estimation accuracy: 1.28 ± 0.06 -> 0.98 ± 0.12 | Helped |
| Use Frugal Flow instead of outcome regression | Causal ATE inference · M1, true ATE 5 | ATE estimation accuracy: 5.29 ± 0.04 -> 5.00 ± 0.24 | Helped |
| Use Frugal Flow instead of outcome regression | Causal ATE inference · M2, true ATE 1 | ATE estimation accuracy: 1.46 ± 0.07 -> 1.01 ± 0.10 | Helped |
| Use Frugal Flow instead of outcome regression | Causal ATE inference · M2, true ATE 5 | ATE estimation accuracy: 5.44 ± 0.05 -> 5.01 ± 0.18 | Helped |
| Use Frugal Flow instead of outcome regression | Causal ATE inference · M3, true ATE 1 | ATE estimation accuracy: 1.13 ± 0.06 -> 1.00 ± 0.09 | Helped |
| Use Frugal Flow instead of outcome regression | Causal ATE inference · M3, true ATE 5 | ATE estimation accuracy: 5.13 ± 0.26 -> 5.18 ± 0.30 | No clear effect |
Surprising findings
- Authors do not recommend using Frugal Flows to statistically infer treatment effects on real-world data, since they need large datasets and causal benchmarks are usually small.
- On the 401(k) benchmark with randomisation-like structure, all inference methods including difference of means recover the correct ATE, as the propensity flow mainly adds outcome noise.
Limitations
- Requires large datasets to infer causal margins accurately; performs poorly on small samples
- Extensive hyperparameter tuning of normalising flows is needed and may cause copula overfitting
- Dequantisation of discrete covariates implicitly imposes ordering, making it less suitable for categorical data
Experimental setup
Datasets: Lalonde (LaLonde jobs dataset) 401(k) eligibility dataset simulated causl-generated data (models M1, M2, M3)
Benchmarks: simulated inference models M1, M2, M3 Lalonde 401(k)
Metrics: ATE estimate mean and 2-sigma confidence interval bias relative to true ATE
Baselines: outcome regression propensity score matching causal normalising flows
Builds on: frugal parameterisation copula flows neural spline flows causal normalising flows marginal structural models
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