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Distinguishing Cause from Effect with Causal Velocity Models
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Proposes causal velocity models for bivariate SCMs, using a score-based goodness-of-fit objective that regresses the velocity against the score for simulation-free causal direction discovery without noise distribution assumptions.
Problem
Bivariate causal direction discovery is limited by functional model classes (ANM, LSNM) that may be misspecified; likelihood methods require noise distribution specification.
Approach
Parametrize bivariate SCMs by a causal velocity (viewing the cause as time in an ODE), show the velocity is tied to the joint score via the continuity equation, and use a score-based goodness-of-fit objective in both directions to pick the causal direction, avoiding noise distribution assumptions and simulation.
Key results
- On the synthetic Velocity benchmark, B-QUAD achieves 97% accuracy (98 AUDRC) versus at most 66% for the best baseline (IGCI).
- On SIM-G with the KDE score estimator, B-QUAD reaches 94% accuracy (99 AUDRC), the best among compared methods on that benchmark.
- With true Gaussian-noise scores, the causal direction is identified with certainty even with n=100 samples.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a quadratic basis velocity instead of LOCI (HSIC) | Causal velocity models · Velocity synthetic | accuracy: 40 -> 97 | Helped |
| Use a linear basis velocity instead of LOCI (HSIC) | Causal velocity models · Sigmoid synthetic | accuracy: 70 -> 87 | Helped |
| Use an ANM velocity parametrization instead of LOCI (HSIC) | Causal velocity models · ANM synthetic | accuracy: 97 -> 92 | Hurt |
| Use an LSNM velocity parametrization instead of LOCI (HSIC) | Causal velocity models · LSNM synthetic | accuracy: 83 -> 82 | No clear effect |
| Use KDE rather than Stein score estimation | Causal velocity models · SIM-G | accuracy: 71 -> 94 | Helped |
| Use KDE rather than Stein score estimation | Causal velocity models · Tübingen | accuracy: 54 -> 71 | Helped |
| Restrict the Tübingen benchmark to continuous observations | Causal velocity models · Tübingen | accuracy: 71 -> 78 | Mixed |
| Provide true scores rather than estimated scores | Causal velocity models · Synthetic ANM and LSNM | causal discovery accuracy | Helped |
Surprising findings
- Taking only the continuous subset of the Tübingen pairs improves the velocity method's performance while existing methods do not improve.
- Using the KDE score estimate improves causal discovery on Gaussian-noise SIM-G and on the real Tübingen benchmark.
Limitations
- Accurate score estimation is required; poor finite-sample score estimates degrade causal discovery.
- Method requires existence of log-densities, so integer-valued data are problematic.
- General identifiability analysis for velocity models remains open.
Experimental setup
Datasets: SIM SIM-C SIM-G Tübingen cause-effect pairs Tübingen (continuous subset) synthetic Velocity, Sigmoid, ANM, LSNM benchmarks
Benchmarks: SIM SIM-C SIM-G Tübingen cause-effect pairs Velocity Sigmoid ANM LSNM
Metrics: accuracy AUDRC
Baselines: LOCI (HSIC) LOCI (Lik) CDS IGCI RECI CGCI
Builds on: RESIT LOCI Stein gradient estimator flow matching ANM LSNM
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