Research map / Probabilistic and scientific modeling / Treatment effects
Conformal Inference of Individual Treatment Effects Using Conditional Density Estimates
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Proposes a two-stage weighted conformal framework for individual treatment effect intervals that uses reference-distribution-estimated conditional densities as conformity scores, yielding shorter intervals than WCP and CM baselines.
Problem
Constructing valid but not overly conservative prediction intervals for individual treatment effects (ITE) under the potential outcome framework, where counterfactual outcomes are unobserved.
Approach
Use the estimated conditional density f(y|x) as the conformity score (shown optimal for shortest interval) and estimate it efficiently via a reference-distribution classification trick, plugged into a two-stage weighted conformal ITE framework (CD-Exact, CD-Naive, plus heuristic CD-Inexact and CD-X).
Key results
- On NSLM, CD-Inexact achieves 92.0% coverage with average length 2.09 versus 93.6%/2.20 for WCP-Inexact; CD-Exact averages 4.13 vs 4.48 for WCP-Exact.
- On ACIC, CD-X gives the shortest intervals (12.2 average length, 93.9% coverage), versus 16.6 for WCP-Inexact.
- On IHDP, CD-Naive yields the shortest guaranteed interval at 20.2 average length with 96.6% coverage, versus 21.4 for WCP-Naive and 82.1 for CM-DR.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use conditional density estimates instead of WCP scores, keeping the paired interval procedure fixed | CD methods versus WCP methods · Semi-synthetic benchmarks: NSLM, ACIC, IHDP | Avg. len. | Helped |
| Use CD-Inexact instead of WCP-Inexact | Inexact ITE intervals · Simulation scenarios | Avg. interval length: 0.26 shorter | Helped |
| Use CD-Exact instead of WCP-Exact | Exact ITE intervals · Simulation scenarios | Avg. interval length: 1.43 shorter | Helped |
| Use CD-Naive instead of WCP-Naive | Naive ITE intervals · Simulation scenarios | Avg. interval length | No clear effect |
| Use CD-X, the propensity-weighted X-learner-style interval method | CD framework · ACIC | Avg. len. | Helped |
| Use CD-Inexact | CD framework · NSLM | Avg. len. | Helped |
| Use CD-Exact | CD framework · Semi-synthetic benchmarks | Coverage: 99.9% | Mixed |
| Use CD-X rather than a method with a coverage guarantee | CD framework · Semi-synthetic benchmarks | Coverage | Hurt |
Surprising findings
- CD-Exact consistently exceeds 99% coverage despite a 90% target, showing that counterfactual uncertainty makes the exact two-stage methods very conservative.
- Estimating propensity scores for CM-DR on IHDP (imbalanced, 139 treated vs 608 control) gives far wider intervals than the 16.7 reported in prior work with known propensity.
Limitations
- CD-X and CD-Inexact lack theoretical coverage guarantees.
- CD-Exact and CD-Naive remain overly conservative (over 99% coverage), with stage-two conservatism unresolved.
- Evaluation relies on simulations and semi-synthetic data since counterfactuals are unobservable.
Experimental setup
Datasets: NSLM ACIC 2016 IHDP simulated data
Benchmarks: NSLM ACIC 2016 IHDP simulation (four scenarios)
Metrics: empirical coverage average prediction interval length
Baselines: WCP-Inexact WCP-Exact WCP-Naive CM-DR
Builds on: weighted conformal prediction conformal meta-learners X-learner reference distribution density estimation
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