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Bivariate Causal Discovery with Proxy Variables: Integral Solving and Beyond

ICML 2025 · Probabilistic and scientific modeling › Causal discovery · approach: Identifiability with hidden confounders

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Proposes a non-parametric kernel-based test (PMCR) for bivariate causal discovery with a negative control outcome via the integral equation, shows the integral equation can have solutions under the alternative when NCO strongly affects Y, and restores identifiability with an additional negative control exposure.

Problem

Bivariate causal discovery under unmeasured confounding using proxy variables: existing integral-equation tests either assume parametric models or discretize continuous variables, and the assumption that the integral equation fails under the causal alternative is unclear and can break when the negat

Approach

Replace first-moment conditional moment restrictions with a proxy maximum characteristic restriction (characteristic-function-based moment conditions solved in an RKHS via Tikhonov regularization) to test the integral equation without discretization; when NCO affects Y, add a second integral restriction using a negative control exposure Z to recover identifiability.

Key results

• In the single-proxy setting, the proposed test controls type-I error near 0.05 while Liu et al. and KCI fail, and its power approaches one as sample size grows.
• On the example where first-moment MMR has no power under H1, PMCR power approaches 1 while MMR stays near alpha=0.05.
• In the two-proxy setting with gamma_W=1, the single-proxy test has low power, while adding the negative control exposure Z raises power to approach one as n increases.

Measured effects reported in the paper

Surprising findings

• When the negative control outcome has a strong direct effect on the outcome, the integral equation can have a solution even under the causal alternative, so single-proxy methods lose power and become non-identifiable.

Limitations

• Analysis only covers the case where the NCO has a direct effect on the outcome; bidirectional edge between NCO and outcome is not analyzed.
• Uniform consistency is impossible for conditional equation testing, so smoothness (RKHS) assumptions are needed.
• Experiments are limited to synthetic data.

Experimental setup

Datasets: synthetic data generated following Liu et al. 2023

Benchmarks: synthetic data (single-proxy and two-proxy settings)

Metrics: type-I error rate power

Baselines: Liu et al. 2023 KCI Miao et al. 2018 MMR (first-moment restriction)

Builds on: Liu et al. 2023 proximal causal discovery Miao et al. 2023 integral equation test Miao et al. 2018 proxy variables Maximum Moment Restriction (Mastouri et al. 2021) KCI

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