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Conformal Prediction for Causal Effects of Continuous Treatments
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Proposes conformal prediction intervals with finite-sample coverage for potential outcomes of continuous treatments under known or estimated (unknown) propensity scores, via dual/KKT-based calibration under propensity shift.
Problem
Conformal prediction for potential outcomes and causal effects of continuous treatments is missing: existing causal CP methods assume known propensity or handle only binary/discrete treatments, and interventions induce covariate shift that breaks exchangeability.
Approach
Reformulates split conformal prediction as augmented quantile regression and calibrates non-conformity scores conditionally on a propensity-shift tilting function (a kernel-smoothed Dirac delta for hard interventions), with a bounded propensity-estimation-error assumption yielding finite-sample coverage even when the propensity is estimated.
Key results
- On synthetic data with hard interventions a*=7x and 10x, the proposed CP intervals reach coverage at or above 1-alpha (e.g., 0.90 at alpha=0.1 for dataset 1 with a*=7x), while MC dropout achieves only 0.13-0.28 coverage at the same levels.
- On semi-synthetic TCGA, CP coverage at alpha=0.05 ranges 0.94-0.97 across interventions, whereas MC dropout gives 0.79-0.86 and the ensemble 0.06-0.09.
- Compared with a weighted CP method for binarized treatments, the proposed intervals are narrower (average width 0.6255 vs. 3.2876) while maintaining coverage.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use conformal intervals instead of MC dropout | MLP potential-outcome predictions · Synthetic Dataset 1, hard interventions a*=7x and a*=10x | empirical coverage: 1.00 vs 0.19; 0.90 vs 0.13; 0.83 vs 0.11; 1.00 vs 0.28; 0.91 vs 0.23; 0.88 vs 0.11 | Helped |
| Use conformal intervals instead of MC dropout | MLP potential-outcome predictions · Synthetic Dataset 2, hard interventions a*=7x and a*=10x | empirical coverage: 1.00 vs 0.02; 0.94 vs 0.02; 0.85 vs 0.02; 1.00 vs 0.08; 0.84 vs 0.07; 0.83 vs 0.07 | Helped |
| Use GP intervals under a large distribution shift | Gaussian process regression · Synthetic Dataset 2, hard intervention | empirical coverage: 0.125; 0.125; 0.0833 | Hurt |
| Use conformal prediction rather than binarizing the continuous treatment and applying binary-treatment CP | Causal prediction intervals · TCGA | interval width: 0.6255 (sd = 0.1714) vs 3.2876 (sd = 0.5587) | Helped |
Surprising findings
- Deep ensemble intervals achieve very low coverage (around 0.05-0.09 at alpha=0.05 on TCGA), indicating they are highly unfaithful.
- The Gaussian process baseline achieves very low empirical coverage (0.0833-0.125) on dataset 2 after intervention, despite small intervals.
Limitations
- Relies on quality of propensity estimator and requires practitioner-specified bound M on estimation error; poor estimates may yield wide intervals.
- Intervals are conservative for point interventions and for output regions with limited calibration data.
- Optimization-based computation can be expensive for large calibration sets and large CATE vectors.
Experimental setup
Datasets: synthetic datasets (2) MIMIC-III TCGA
Benchmarks: synthetic dataset 1 synthetic dataset 2 MIMIC-III TCGA (semi-synthetic)
Metrics: empirical coverage interval width
Baselines: MC dropout deep ensemble Gaussian process regression vanilla conformal prediction weighted conformal method for binary treatments
Builds on: split conformal prediction conformal prediction under covariate shift conformal causal inference for binary treatments
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