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Tight Partial Identification of Causal Effects with Marginal Distribution of Unmeasured Confounders

ICML 2024 · Probabilistic and scientific modeling › Causal discovery · approach: Causal effect identification

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Derives closed-form tight partial identification bounds for causal effects in a binary-treatment/outcome, discrete-confounder model using only the marginal distribution P(U), and gives if-and-only-if conditions for when this information yields non-vanilla bounds.

Problem

Partial identification of interventional probabilities and ATE under unmeasured confounding when only the marginal distribution P(U) of confounders is known, without extra restrictions (entropy, auxiliary variables) on P(U).

Approach

Derives the tight PI region in closed form for any P(U) in the discrete case, via piecewise-monotone analysis and subset-sum characterization, and proves that non-vanilla bounds occur iff P(U) falls outside an interval-based set whose probability becomes less likely as confounder cardinality grows.

Key results

• Non-vanilla PI probability (P(U) outside the degeneration set) decays toward zero as confounder cardinality d_u grows; reported to be below 0.1 for d_u >= 10 regardless of P(X,Y).
• Closed-form tight bounds for P(y|do(x)) are derived for binary U and generalized to any discrete U via if-and-only-if conditions (Theorems 4.1, 4.2).
• Proposed bound for ATE validly controls the gap to the vanilla bound and is reported to outperform separately estimated bounds of Jiang et al. (2023) on INSURANCE and ADULT.

Measured effects reported in the paper

Surprising findings

• Li et al. (2023)'s entropy-based bound is looser than the vanilla bound whenever P(x) > P(not x) and epsilon lies in a certain range, so it can be non-informative.
• The if-and-only-if degeneration conditions for ATE are consistent with those for the interventional probability in the multi-valued case, contrary to the binary-case intuition.

Limitations

• Analysis restricted to the simplest causal graph (single confounder U, binary X and Y) and discrete U.
• Multi-valued ATE extension and non-closed-form subset-sum quantities are only approximated.
• Negative result on cardinality is based on uniform sampling simulations and bounds that are not necessarily tight.

Experimental setup

Datasets: INSURANCE (Binder et al., 1997) ADULT (Dua & Graff, 2017)

Benchmarks: INSURANCE ADULT

Metrics: width of identification region fraction of non-vanilla P(U) cases probability of non-vanilla region vs. confounder cardinality

Baselines: Vanilla bound (Tian & Pearl, 2000) Li et al. 2023 Jiang et al. 2023

Builds on: Tian & Pearl vanilla bounds Li et al. 2023 closed-form PI Jiang et al. 2023 entropy-based PI Duarte et al. 2023 branch-and-bound PI

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