Research map / Probabilistic and scientific modeling / Variational and generative sampling
Exogenous Matching: Learning Good Proposals for Tractable Counterfactual Estimation
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Proposes Exogenous Matching (EXOM), an importance-sampling method for counterfactual probability estimation that turns variance minimization into conditional exogenous-distribution learning, evaluated on SCMs and identifiable proxy SCMs.
Problem
Tractable and efficient estimation of counterfactual (L3) expressions in general settings, where exogenous variables may be discrete or continuous and naive Monte Carlo/rejection sampling is inefficient for rare counterfactual events.
Approach
Derives a common variance upper bound for conditional importance proposals, reducing it to maximizing expected log-density of the exogenous distribution at potential responses (cross-entropy-like objective), so a single conditional proposal q(u|y*) is learned and reused across counterfactual events; injects counterfactual Markov boundaries as weight masks into the conditioning network.
Key results
- On SIMPSON-NLIN with |s|=1, EXOM[MAF] reaches ESP 0.581 and FR 0.005, versus 0.008 ESP for rejection sampling (FR 0.971).
- At |s|=3 on SIMPSON-NLIN, rejection sampling, CEIS and NIS all have ESP 0.000 and FR 1.000, while EXOM[MAF] reaches ESP 0.606 and FR 0.075.
- On proxy SCMs, EXOM with CausalNF on SIMPSON-NLIN yields FR 0.40 at |s|=3 and 0.61 at |s|=5, while RS fails (FR 1.0); FAIRNESS bias estimates with NCM are near 0.01 for both RS and EXOM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use EXOM with MAF instead of the strongest listed baseline | Counterfactual importance sampling · SIMPSON-NLIN (|s|=1,3,5) | ESP: 0.037 -> 0.581 (|s|=1); 0.000 -> 0.606 (|s|=3); 0.000 -> 0.698 (|s|=5) | Helped |
| Use EXOM with MAF instead of the strongest listed baseline | Counterfactual importance sampling · NAPKIN (|s|=1,3,5) | ESP: 0.017 -> 0.306 (|s|=1); 0.000 -> 0.397 (|s|=3); 0.000 -> 0.482 (|s|=5) | Helped |
| Use EXOM with MAF instead of the strongest listed baseline | Counterfactual importance sampling · FAIRNESS-XW (|s|=1,3,5) | ESP: 0.193 -> 0.339 (|s|=1); 0.122 -> 0.261 (|s|=3); 0.106 -> 0.237 (|s|=5) | Helped |
| Use EXOM with MAF instead of the strongest listed baseline | Counterfactual importance sampling · SIMPSON-NLIN, NAPKIN, and FAIRNESS-XW | FR: See Table 1 | Helped |
| Add counterfactual Markov-boundary masks to conditioning (ablation) | EXOM · SIMPSON-NLIN, LARGEBD-NLIN, M, and NAPKIN | ESP | Helped |
| Use MAF rather than GMM as the conditional density model (ablation) | EXOM · SIMPSON-NLIN, NAPKIN, and FAIRNESS-XW | ESP | Mixed |
Surprising findings
- On the FAIRNESS-XW discrete SCM, EXOM[GMM] attains higher ESP than EXOM[MAF] at |s|=1 (0.358 vs 0.339), contrary to the expectation that MAF dominates.
Limitations
- Requires a partially specified SCM (black-box mechanisms and sampleable exogenous distribution), relying on identifiable proxy SCMs that are still an active research area.
- The general case assumes finite submodels and approximability by a finite number of counterfactual probabilities.
- Markov boundary masking and related tricks may not transfer to SCMs with general parameter settings.
Experimental setup
Datasets: SIMPSON-NLIN TRIANGLE-NLIN LARGEBD-NLIN M NAPKIN FAIRNESS FAIRNESS-XW
Benchmarks: SIMPSON-NLIN NAPKIN FAIRNESS-XW FAIRNESS
Metrics: ESP FR average bias with 95% CI
Baselines: Rejection Sampling CEIS NIS
Builds on: neural importance sampling cross-entropy importance sampling multiple importance sampling causal normalizing flows
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