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Instrumental and Proximal Causal Inference with Gaussian Processes

Yuqi Zhang, Krikamol Muandet, Dino Sejdinovic, Edwin Fong, Siu Lun Chau

Abstract

Instrumental variable (IV) and proximal causal learning (Proxy) methods are central frameworks for causal inference in the presence of unobserved confounding. Despite substantial methodological advances, existing approaches rarely provide reliable epistemic uncertainty (EU) quantification. We address this gap through a Deconditional Gaussian Process (DGP) framework for uncertainty-aware causal learning. Our formulation recovers popular kernel estimators as the posterior mean, ensuring predictive precision, while the posterior variance yields principled and well-calibrated EU. Moreover, the probabilistic structure enables systematic model selection via marginal log-likelihood optimization. Empirical results demonstrate strong predictive performance alongside informative EU quantification, evaluated via empirical coverage frequencies and decision-aware accuracy rejection curves. Together, our approach provides a unified, practical solution for causal inference under unobserved confounding with reliable uncertainty.

Instrumental and Proximal Causal Inference with Gaussian Processes

Abstract

Instrumental variable (IV) and proximal causal learning (Proxy) methods are central frameworks for causal inference in the presence of unobserved confounding. Despite substantial methodological advances, existing approaches rarely provide reliable epistemic uncertainty (EU) quantification. We address this gap through a Deconditional Gaussian Process (DGP) framework for uncertainty-aware causal learning. Our formulation recovers popular kernel estimators as the posterior mean, ensuring predictive precision, while the posterior variance yields principled and well-calibrated EU. Moreover, the probabilistic structure enables systematic model selection via marginal log-likelihood optimization. Empirical results demonstrate strong predictive performance alongside informative EU quantification, evaluated via empirical coverage frequencies and decision-aware accuracy rejection curves. Together, our approach provides a unified, practical solution for causal inference under unobserved confounding with reliable uncertainty.
Paper Structure (49 sections, 15 theorems, 85 equations, 15 figures, 6 tables)

This paper contains 49 sections, 15 theorems, 85 equations, 15 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Preliminary
  4. Notations.
  5. Instrumental Variable
  6. Proximal Causal Learning
  7. Kernel embeddings of distributions
  8. Kernel mean embeddings (KME).
  9. Conditional mean embeddings (CME).
  10. Deconditional mean embeddings (DME).
  11. Methods: Deconditional GP for IV and Proxy
  12. GP models for IV
  13. GP models for Proxy
  14. Hyperparameter Selection
  15. Related Works
  16. ...and 34 more sections

Key Result

Proposition 3.1

(CMP for IV.) Under assumptions assumptioniv1, assumptioniv2, the conditional mean process $\{g(z),z \in {\mathcal{Z}}\}$ induced by $f$ with respect to $P_{X\mid Z}$ defined as: is a Gaussian Process $g \sim {\mathcal{G}}{\mathcal{P}}(0, q)$, with covariance kernel where $\mu_{X\mid Z=z}$ is the CME of $P_{X\mid Z=z}$ in the RKHS ${\mathcal{H}}_{\mathcal{X}}$.

Figures (15)

  • Figure 1: Causal Graphs
  • Figure 2: Accuracy-rejection curve for log design (data size $=200$), with quantile $= 0.65,0.75,0.85$
  • Figure 3: DAG of Proxy setting when the observed confounders exist.
  • Figure 4: Demo of synthetic data with data size $= 1000$. Left: log, middle: linear, right:sine.
  • Figure 5: Accuracy-rejection curve for sine design (data size $=200$), with quantile $= 0.65,0.75,0.85$.
  • ...and 10 more figures

Theorems & Definitions (25)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Proposition C.3
  • proof
  • Proposition C.4
  • ...and 15 more