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Conditional Common Entropy for Instrumental Variable Testing and Partial Identification
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Introduces conditional common entropy (CCE) to bound causal effects with weakly invalid instrumental variables under an entropy bound on the confounder, and to falsify invalid IVs, with an iterative approximation algorithm.
Problem
Causal effects with instrumental variables are often not identifiable, and IV validity (exclusion and exchangeability) is generally untestable from observational data.
Approach
Defines conditional common entropy, the minimum entropy of a latent variable that renders treatment-outcome-instrument conditionally independent, and uses it as a testable necessary condition for valid IVs and as a sensitivity parameter in partial identification bounds under a weak-confounding (bounded confounder entropy) assumption.
Key results
- On the lung cancer case study, CCE_IV = 0.393 exceeds the confounder entropy bound θ = 0.31, allowing rejection of the invalid IV, and the entropy-constrained ACE bounds tighten from [-0.272, 0.828] to [-0.078, 0.828].
- On PimaIndiansDiabetes (binarized), CCE_IV is 0.66 for insulin versus 0.91 for BMI, suggesting insulin is the more likely valid instrument for glucose on blood pressure.
- Synthetic experiments show that for high-dimensional variables the valid IV almost always attains smaller CCE_IV than an invalid IV with a direct effect on Y.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply the entropy constraint | Proposed causal-effect bounds · Lung Cancer dataset | ACE lower bound: −0.272 -> −0.078 | Helped |
| Use the graph-specific conditional common entropy criterion | Smoking as an instrument for bronchitis → dyspnoea · Lung Cancer dataset | IV falsification: 0.104 -> 0.393 | Helped |
| Choose insulin rather than BMI using the lower CCEIV | Instrument selection for glucose → blood pressure · PimaIndiansDiabetes | CCEIV: 0.66 vs 0.91 | Helped |
| Select the candidate with smaller CCEIV | Instrument selection heuristic · Synthetic data with valid Z and invalid V | CCEIV | Helped |
Surprising findings
- CCE does not change monotonically with the entropy of the latent confounder, since a high-entropy confounder may induce only a weak connection.
Limitations
- Requires prior knowledge of an upper bound on confounder entropy (weak confounding assumption) for bounds and falsification.
- Test can only falsify invalid instruments; it is a necessary, not sufficient, condition for validity.
- Empirical real-data examples require discretizing continuous variables and rely on a synthetic lung cancer dataset.
Experimental setup
Datasets: synthetic data sampled from IV graph factorization lung cancer synthetic dataset (Lauritzen & Spiegelhalter, 1988) PimaIndiansDiabetes dataset (Smith & Dickson, 1988) hypothetical drug trial example
Benchmarks: lung cancer dataset (bronchitis-dyspnoea) PimaIndiansDiabetes (glucose-blood pressure) synthetic IV graphs
Metrics: CCE approximation vs. confounder entropy bound gap CCE for IV selection
Baselines: natural bounds Tian-Pearl bounds entropic partial identification (Jiang et al., 2023) Cinelli & Hazlett sensitivity analysis
Builds on: entropic causal inference (common entropy) entropic partial identification natural bounds (Robins/Manski) Tian-Pearl bounds single world intervention graphs
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