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Probabilities of Causation with Nonbinary Treatment and Effect
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Derives sharp and recursive bounds for probabilities of causation with nonbinary treatment and effect using experimental and observational data, illustrated with examples and simulations.
Problem
Estimating probabilities of causation (PNS, PN, PS and higher-order variants) when treatment and effect take more than two values, where existing bounds were restricted to binary variables and Balke's linear programming is computationally impractical.
Approach
Extends Tian-Pearl bounds to multivalued treatment and effect by applying Frechet inequalities and the consistency rule to reduce each probability of causation to experimental and observational distributions, giving closed-form sharp bounds for single-hypothetical-term cases and recursive bounds for multi-term cases.
Key results
- Bounds for P(y1x1,y1x2) with m=2, n=3 contain the true value across 1000 random samples, with midpoints close to true values.
- Average bound gap over 1000 simulated samples is 0.228.
- Cancer treatment example gives P(y3x1,y1x2,y2x3) between 0 and 0.099, suggesting radiation should not be chosen.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed bounds with the cancer-treatment experimental and observational data | PNS(3) for surgery, chemotherapy, and radiation · Cancer treatment example | PNS(3) upper bound: 0 ≤P(y3x1, y1x2, y2x3) ≤0.099 | Hurt |
| Choose Institute B instead of Institute A | Bob's job-search institute choice · Institute choice example | P(y1x3|x2,y2) lower bound: 0.720 ≤P(y1x3|x2, y2) ≤1 | Helped |
| Choose Institute C instead of Institute A | Bob's job-search institute choice · Institute choice example | P(y1x4|x2,y2) upper bound: 0 ≤P(y1x4|x2, y2) ≤0.042 | Hurt |
| Vaccinate rather than leave unvaccinated | Vaccine clinical study · Severe condition if unvaccinated; uninfected if vaccinated | P(y1x1,y4x2) upper bound: 0 ≤P(y1x1, y4x2) ≤0.039 | No clear effect |
| Vaccinate rather than leave unvaccinated | Vaccine clinical study · Severe condition if unvaccinated; asymptomatic if vaccinated | P(y2x1,y4x2) upper bound: 0.037 ≤P(y2x1, y4x2) ≤0.077 | No clear effect |
| Vaccinate rather than leave unvaccinated | Vaccine clinical study · Severe condition if unvaccinated; mild symptoms if vaccinated | P(y3x1,y4x2) lower bound: 0.502 ≤P(y3x1, y4x2) ≤0.561 | Helped |
| Apply the proposed theorems to generated observational and experimental samples | Proposed probability-of-causation bounds · P(y1x1,y1x2), 1000 simulated samples | Average bound gap: 0.228 | No clear effect |
Limitations
- Bounds for multi-hypothetical-term probabilities (Theorems 8-11) are not sharp due to recursive construction.
- Bounds are not shown to be tight for general graphical settings; only the no-covariate case is treated.
- Simulation covers only one probability with m=2, n=3.
Experimental setup
Datasets: simulated experimental and observational distributions hypothetical clinical, job-market and vaccine tables
Benchmarks: simulated SCM-generated distributions
Metrics: bound width coverage of true probability of causation
Baselines: Balke linear programming Tian-Pearl bounds
Builds on: Tian-Pearl bounds Balke linear programming probability of necessity and sufficiency
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