Proximal Causal Inference Framework

Updated 25 October 2025

Proximal causal inference is a formal framework that addresses bias from unmeasured confounding by using proxy variables to indirectly account for latent confounders.
It introduces novel identification conditions and confounding bridge functions that generalize classical g-computation and instrumental variable techniques.
The framework supports robust, semiparametric estimation through methods like proximal 2SLS, inverse probability weighting, and doubly robust estimators.

Proximal causal inference is a formal statistical framework designed to recover causal effects from observational data when the traditional assumption of no unmeasured confounding (exchangeability) is violated, but observed covariates serve as imperfect proxies for the true latent confounders. Rather than conditioning on observed covariates to achieve exchangeability, this paradigm explicitly acknowledges that available covariates may only partially reflect underlying confounding mechanisms and leverages their proxy structure to identify and estimate causal parameters. Central contributions include the formulation of new identification conditions, nonparametric and semiparametric estimation theory, the proximal g-formula, and a suite of practical and efficient estimation algorithms, all rigorously justified with formal completeness and model-bridge assumptions. This framework generalizes and extends classical g-computation, doubly robust estimation, and instrumental variable techniques, allowing causal learning in settings where prior methods fail due to unaddressed confounding bias.

1. Conceptual Foundations and Motivating Weaknesses of Exchangeability

Traditional causal inference in observational studies presumes that a sufficiently rich set of measured covariates $L$ has been collected so that, upon conditioning, the treatment $A$ is independent of all potential outcomes $Y_a$ (the so-called exchangeability or ignorability assumption). This assumption is often implausible in real-world applications, as unmeasured variables $U$ may influence both treatment and outcome. Proximal causal inference directly confronts this skepticism. It reframes the observed covariates $L$ as consisting of:

Type a: true common causes (measured confounders),
Type b: treatment-inducing confounding proxies, correlated with treatment via the latent $U$ ,
Type c: outcome-inducing confounding proxies, correlated with outcome via $U$ .

This explicit categorization recognizes that even a rich set of measured variables fails to guarantee exchangeability and that, instead, the proxies' association structures must be leveraged to control for unmeasured confounding indirectly (Tchetgen et al., 2020, Cui et al., 2020).

2. Formal Identification Theory: Proxy Structure and Completeness

Because measured proxies do not fully capture unmeasured confounding, proximal identification replaces conditional exchangeability with alternative sufficient conditions. A critical concept is completeness, formalized as:

$E\{v(U) | z, a, x\} = 0 \,\text{for all }(z,a,x) \Longrightarrow v(U)=0,$

where $v(U)$ ranges over square-integrable functions of the latent confounder. This condition, or its observable analogs, guarantees that the distribution of proxies contains enough information about $U$ so that key integral equations (of Fredholm type) relating observed and latent data are invertible (Tchetgen et al., 2020, Cui et al., 2020).

In categorical settings, completeness is operationalized by requiring that the support of proxies $Z$ and $W$ be at least as large as that of $U$ . If this fails, point identification may not be possible, motivating partial identification approaches using bounds (Ghassami et al., 2023).

3. Core Methodology: The Proximal g-Formula and g-Computation Algorithm

The proximal g-formula is a generalization of Robins' foundational g-formula that "bridges" from the observed proxies to the latent exchangeability structure. Let $Z$ be the set of treatment-inducing proxies and $W$ outcome-inducing proxies, with baseline covariates $X$ . The key identification step is to find a "confounding bridge function" $h(a,x,w)$ solving:

$E(Y \mid a,z,x) = \sum_w h(a,x,w) f(w \mid a,z,x).$

Once $h(\cdot)$ is established (under completeness), the average potential outcome under treatment $a$ is given by the proximal g-formula:

$\beta(a) = \sum_{w,x} h(a,x,w) f(w,x).$

Estimation proceeds by modeling (parametrically or semiparametrically) the conditional distribution $f(w \mid a,z,x)$ and $h(\cdot)$ . In practice, a two-stage procedure (proximal 2SLS or g-computation) is employed:

Regress $W$ on $(A,Z,X)$ to form predicted proxy scores,
Regress $Y$ on $(A,X,\hat{W})$ , yielding consistent estimators for the causal effect of interest, even in the presence of unmeasured confounding (Tchetgen et al., 2020, Cui et al., 2020).

4. Semiparametric Efficiency, Doubly Robust, and Minimax Estimators

Proximal causal inference is closely tied to the development of efficient and robust estimation methods. Under the semiparametric model, the efficient influence function (EIF) for the average treatment effect is:

$\operatorname{EIF}(\psi) = (-1)^{1-A} q(Z, A, X) [Y - h(W, A, X)] + h(W, 1, X) - h(W, 0, X) - \psi,$

where $q(\cdot)$ is a treatment-bridge function solving a related Fredholm equation (Cui et al., 2020, Ghassami et al., 2021). Proximal estimators include:

The Proximal Outcome Regression (POR) estimator, which averages predictions from $h(\cdot)$ ,
The Proximal Inverse Probability Weighting (PIPW) estimator, weighting by $q(\cdot)$ ,
The Proximal Doubly Robust (PDR) estimator, combining both and achieving consistency if either $h$ or $q$ is correctly specified.

The minimax kernel machine learning framework further extends these ideas by constructing doubly robust estimators in which the bridge functions are solutions to integral equations, optimized over RKHS and regularized to address the ill-posedness of the inverse problem (Ghassami et al., 2021).

5. Extensions: Longitudinal Data, Mediation, and Graphical Models

The proximal framework is generalized to accommodate longitudinal interventions and time-varying confounding, using recursively defined bridge functions subject to sequential completeness (Ying et al., 2021). In mediation settings with unmeasured confounding between exposure, mediator, and outcome, or even with hidden (unmeasured) mediators, proximal bridge functions support identification of natural direct and indirect effects. The resulting proximal mediation analysis yields multiply robust and locally efficient estimators for these effects (Dukes et al., 2021, Ghassami et al., 2021).

Under graphical causal models, the Proximal ID Algorithm synthesizes classical ID-algorithm–based identification with proxy variable strategies. By systematically inserting "proximal fixing" steps—using bridge functions—into valid fixing sequences in a causal graph, the framework achieves identification in all cases where standard ID applies and in broader settings where proxy information is available but unmeasured confounders violate the Markov property (Shpitser et al., 2021).

6. Practical Applications and Empirical Illustrations

Proximal methods have been empirically applied to settings such as the SUPPORT paper assessing right heart catheterization (RHC) on 30-day mortality in ICU patients. These analyses typically:

Select candidate proxies (e.g., pafi1, paco21 as treatment proxies, ph1, hema1 as outcome proxies),
Employ proximal 2SLS estimation,
Compare results to standard OLS or exchangeability-based estimators.

In these examples, proximal estimators yield larger absolute effects (suggesting more severe bias from unmeasured confounding in the naïve analyses), with increased standard errors reflecting the additional uncertainty captured by the proxy-based correction (Tchetgen et al., 2020, Cui et al., 2020). Other empirical domains include longitudinal rheumatoid arthritis studies (Ying et al., 2021) and complex synthetic control settings with latent confounders (Shi et al., 2021, Liu et al., 2023).

7. Challenges, Limitations, and Future Research

Proximal identification rests on assumptions (e.g., correct proxy classification, completeness) that are strong and untestable in practice. Model misspecification or violations of the exclusion restrictions for proxies can render estimates biased; partial identification approaches supplying bounds under weak assumptions offer an alternative (Ghassami et al., 2023). Ongoing research targets:

Proxy selection and validation methods,
Improved diagnostics for completeness,
Extensions to cases with partially invalid proxies or scarce/binary proxies,
Scalable estimation algorithms for high-dimensional or longitudinally complex data,
Minimax and nonparametric kernel-based methods to manage ill-posed inverse problems.

The continual refinement of this theoretical and computational toolkit promises to further enhance the reliability and generalizability of causal inference from imperfectly measured observational data.