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Learning Structural Causal Models from Ordering: Identifiable Flow Models
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Proposes identifiable causal flow models (CFM) that learn SCMs from observational data and a causal ordering via ODE-based triangular monotonic flows, with S-CFM and P-CFM variants for parallel abduction and prediction.
Problem
Learning structural causal models for observational, interventional, and counterfactual queries from only observational data and a causal ordering, with scalable inference as the number of variables grows.
Approach
Model each node's exogenous-to-endogenous map as an ODE flow whose velocity depends only on preceding nodes, so the overall map is a triangular monotonically increasing (TMI) map and is identifiable up to component-wise transforms; a MADE-based velocity network (MAVEN) plus an endogenous predictor allow parallel abduction and prediction in O(n) over layers.
Key results
- P-CFM and S-CFM achieve the best or second-best observational, interventional, and counterfactual MMD/MSE on most of the eight synthetic SCMs versus CausalNF and VACA.
- On the fMRI interventional task, S-CFM reaches median absolute error 0.5829 vs 0.6018 for CausalNF-NSF and 0.6746 for ANM, a slight improvement.
- P-CFM inference time stays nearly constant as the number of nodes grows, while DCM's grows linearly with node count.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use S-CFM instead of CausalNF-NSF | Causal inference model · Triangle NADD | CF. MSE: 41.90_{11.74} -> 6.53_{0.34} | Helped |
| Use S-CFM instead of CausalNF-NSF | Causal inference model · Simpson NLIN | Obs. MMD: 0.62_{0.05} -> 0.57_{0.05} | Helped |
| Use S-CFM instead of CausalNF-NSF | Causal inference model · Diamond NLIN | Int. MMD: 0.34_{0.02} -> 0.41_{0.05} | Hurt |
| Use P-CFM instead of CausalNF-NSF | Causal inference model · Diamond NLIN | CF. MSE: 16.06_{2.07} -> 0.03_{0.01} | Helped |
| Use P-CFM instead of CausalNF-NSF | Causal inference model · Y NADD | Obs. MMD: 0.55_{0.07} -> 0.46_{0.05} | Helped |
| Use S-CFM instead of CAREFL | Causal inference model · fMRI interventional prediction | Median Abs. Error: 0.5983_{3.0\text{e-}2} -> 0.5829_{5.0\text{e-}2} | Helped |
| Replace velocity matching with data denoising objective (ablation) | S-CFM · observational and interventional datasets | MMD | No clear effect |
| Replace velocity matching with data denoising objective (ablation) | S-CFM · counterfactual datasets | MSE | Hurt |
Surprising findings
- CausalNF-MAF, despite being a flow with affine constraints, fails on non-additive SCMs where the endogenous distribution is non-Gaussian.
- The data-denoising (x-prediction) training variant performs nearly as well as velocity matching on observational and interventional metrics but is worse on counterfactuals.
Limitations
- Identifiability proof relies on assumptions: Markovian, acyclic SCMs with diffeomorphic structural equations and a known valid causal ordering.
- On the two-node fMRI task, P-CFM was not evaluated since it shows little advantage in the bivariate case; S-CFM's gain is small and based on a single-intervention absolute error.
- Baseline DCM was not compared in the main experiments because it requires a fully observed graph or expensive parent selection.
Experimental setup
Datasets: synthetic SCMs (Triangle, Simpson, Diamond, Y, LargeBD, LargeLadder) electrical stimulation interventional fMRI data Morpho-MNIST
Benchmarks: synthetic SCMs (Triangle, Simpson, Diamond, Y, LargeBD, LargeLadder) fMRI interventional prediction Morpho-MNIST counterfactuals
Metrics: MMD MSE median absolute error inference time
Baselines: CausalNF-NSF CausalNF-MAF VACA CAREFL ANM Linear SCM DCM
Builds on: CausalNF Rectified Flow Knothe-Rosenblatt transport MADE VACA
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