Exploiting independence constraints for efficient estimation of bounds on causal effects in the presence of unmeasured confounding

Abstract: Causal graphs may inform covariate adjustment for estimating causal effects and improve estimation efficiency by exploiting the graphical structure. In many applications, however, the target causal parameter may not be point-identified due to the presence of unmeasured confounding. Sensitivity analysis methods address this challenge by characterizing bounds on the causal parameter under varying assumptions about the magnitude or form of unmeasured confounding. We focus on semiparametric efficient estimation of causal effects in non-identifiable settings, assuming a known (or hypothesized) causal graph. We propose an influence function projection approach that exploits the conditional independence constraints implied by the graph to improve the efficiency of semiparametric estimators of upper and lower bounds on the average causal effect under a given sensitivity analysis model. Our approach applies across multiple sensitivity analysis frameworks and causal estimands, thereby connecting knowledge of graphical structure with the sensitivity analysis literature. We illustrate our approach through simulations and real data examples thought to be affected by unmeasured confounding, including the effect of labor training program on post-intervention earnings, and the effect of low ejection fraction on heart failure death.

• The paper presents a new projection operator for influence functions that significantly reduces variance in estimating average causal effects under unmeasured confounding.
• It exploits conditional and marginal independence constraints derived from causal graphs, integrating sensitivity analysis to improve estimator efficiency.
• Simulation studies and real data applications, including heart failure mortality and labor training programs, validate the method’s practical efficiency gains.

Efficient Estimation of Causal Bounds via Influence Function Projection Under Unmeasured Confounding

Problem Motivation and Technical Foundations

This paper addresses semiparametric efficient estimation of bounds on causal effects, specifically the average causal effect (ACE), when observational data is subject to unmeasured confounding and hence the causal parameter is not point-identified. The methodology is grounded in sensitivity analysis, which parametrizes (via a sensitivity parameter $\gamma$) the plausible effect of hypothetical unmeasured confounders $U$ on treatment assignment and/or outcome. The approach systematically exploits conditional/marginal independence constraints implied by hypothesized or data-driven causal graphs to increase estimation efficiency. Efficient influence function (IF) projection is used to minimize estimator variance in non-identifiable settings.

The paper comprehensively bridges semiparametric efficiency theory with sensitivity analysis on graphical models. It generalizes previous work on optimal adjustment set selection in point-identified cases [rotnitzky2020efficient] to non-identifiable ones. The proposed projected IFs obey influence function orthogonality corresponding to independence constraints encoded in directed acyclic graphs (DAGs), acyclic directed mixed graphs (ADMGs), or partial ancestral graphs (PAGs).

Main Theoretical Contributions

A new projection operator for IFs is formally derived. Given an IF $\phi(Z)$ and marginal/conditional independence $X_i \perp X_j \mid X_S$, the efficient projected IF is:

$$\phi^\text{eff}(Z) = \phi(Z) - E[\phi(Z)|X_i,X_j,X_S] + E[\phi(Z)|X_i,X_S] + E[\phi(Z)|X_j,X_S] - E[\phi(Z)|X_S]$$

Generalization for multiple independence constraints is provided via an alternating projection algorithm, iteratively applying projection operators to converge to the intersection of tangent spaces. When constraints correspond to a DAG factorization, tangent spaces are orthogonal and one pass suffices; otherwise, e.g. ADMGs, iteration improves convergence.

Critical variance reduction guarantees are established: the difference in asymptotic variance is equal to

$$\text{Var}(E[\phi(Z)|X_i,X_j]) - \text{Var}(E[\phi(Z)|X_i]) - \text{Var}(E[\phi(Z)|X_j])$$

for $X_S = \emptyset$, quantifying the efficiency gain from joint independence structure.

Sensitivity Analysis Models and Semiparametric Estimation

The influence function projection schema is instantiated in several sensitivity analysis frameworks. Primary focus is the exponential tilting model [robins2000sensitivity], parameterizing deviation from ignorability through $\gamma$. For binary $Y$,

$$p(y(t) \mid 1-t, x) = p(y(t) \mid t, x) \frac{\exp(\gamma y)}{E[\exp(\gamma Y) \mid T=t,X=x]}$$

The ACE is estimated nonparametrically for each $U$0 via efficient IFs, and cross-fitted one-step estimators using super learner nuisance function fits. Projection onto independence submodels is shown to lower estimator variance, with proven asymptotic optimality.

The method generalizes to other frameworks, e.g., omitted variable bias bounding [chernozhukov2022long], supporting both binary and continuous treatments and outcomes.

Simulation Evidence

Two key simulation experiments are reported:

• DAG-Faithful Independence Constraints: With known DAG, projection onto correct independence submodels (single iteration) yields substantial variance reduction for ACE estimates across sensitivity parameter values ($U$1), without altering point estimates.
• ADMG/Non-DAG Constraints: Where independence constraints do not correspond to a unique DAG (latent confounding across covariates), iterative projection is necessary. Projected IFs lead to $U$2–$U$3 variance reduction (for $U$4), depending on $U$5, without bias.

Incorrect submodel specification (projecting onto false constraints) is shown to reduce variance but induce severe bias in the estimator, emphasizing the necessity of correct structural assumptions.

Real Data Application: Heart Failure Mortality

The method is applied to the effect of low ejection fraction (EF $U$6) on heart failure mortality, with baseline covariates and possible latent confounding. Marginal independence constraints are learned via tiered PC algorithms. Projecting IFs onto learned constraints significantly narrows ACE confidence intervals for a broad range of $U$7, preserving point estimates. The projected estimator demonstrates strong practical efficiency gains.

Figure 1: Estimated effect of low ejection fraction (EF) on heart failure death as a function of sensitivity parameter $U$8, showing $U$9 confidence intervals for both nonparametric and projected IFs.

Real Data Application: Labor Training Programs

A similar analysis is conducted for the effect of a labor training program (NSW) on post-intervention earnings (binary outcome). Independence constraints among pre-intervention covariates are discovered using the PC algorithm. Cross-fitted super learner nuisance models and projected IFs again narrow confidence intervals for the ACE estimates over multiple $\phi(Z)$0, with invariant point estimates.

Figure 2: Estimated effect of labor training program on post-intervention earnings as a function of sensitivity parameter $\phi(Z)$1, comparing $\phi(Z)$2 confidence intervals from nonparametric and projected IFs.

Implications and Future Directions

The technique offers a general, practical approach for variance reduction in sensitivity-bound causal effect estimation, applicable across diverse sensitivity models and causal estimands. It is robust to non-identifiability due to unmeasured confounding, provided the independence structure is correctly specified.

Post-selection inference issues arise when algorithmic structure learning is used (e.g., learning independence constraints from the same dataset), potentially compromising nominal confidence interval coverage. Integration with post-selection inference frameworks [chang2026post] is recommended.

Theoretical extension to optimal adjustment set discovery in non-identified settings and practical adaptation to continuous outcomes/treatments is warranted. Algorithmic scalability for high-dimensional covariate spaces and more sophisticated independence structure learning will be necessary for broader deployment.

Conclusion

Efficient projection of influence functions onto independence-constrained submodels delivers robust variance reduction in ACE bounds estimation under unmeasured confounding. The method generalizes across sensitivity analysis models, is validated both numerically and in real-world datasets, and expands the arsenal of semiparametric causal inference with strong practical and theoretical implications for future developments in causal machine learning and statistical AI.