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Causal Inference on Networks under Misspecified Exposure Mappings: A Partial Identification Framework

This paper proposes a novel partial identification framework that derives sharp bounds on direct and spillover effects to assess the robustness of causal inference in networks when the exposure mapping is misspecified, offering valid and efficient estimators for three canonical settings.

Original authors: Maresa Schröder, Miruna Oprescu, Stefan Feuerriegel, Nathan Kallus

Published 2026-02-04

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CC BY 4.0

Original authors: Maresa Schröder, Miruna Oprescu, Stefan Feuerriegel, Nathan Kallus

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out if a new vaccine works. In a simple world, you just give the vaccine to some people and see if they get sick. But in the real world, people talk to each other. If your friend gets vaccinated, they might protect you too, even if you didn't get the shot. This is called a network effect or "spillover."

The problem is that we don't always know exactly how that protection spreads. Does it spread to everyone you know? Just your best friends? Does it fade after two degrees of separation?

To study this, scientists usually make a guess (an "exposure mapping") about how the network works. They might say, "Okay, let's assume your risk depends only on the average number of vaccinated friends you have."

The Paper's Big Problem:
What if that guess is wrong? What if the protection actually comes from having one very close vaccinated friend, not the average? If the scientists' guess is wrong, their calculations for how well the vaccine works will be completely off. They might think it works great when it doesn't, or vice versa.

The Paper's Solution: "Partial Identification"
Instead of pretending they know the exact truth (which they don't), the authors propose a new framework that says: "We don't know the exact rule, but we know the rule isn't too far off from our guess. So, let's calculate the best-case and worst-case scenarios."

Think of it like checking the weather.

The Old Way: The meteorologist says, "It will be exactly 72°F." If they are wrong, you get soaked or freeze.
This Paper's Way: The meteorologist says, "We aren't 100% sure of the model, but we know it's likely between 65°F and 75°F." Even if the exact temperature is 73°F, you know you need a light jacket, not a heavy coat. You get a safe range instead of a risky guess.

How They Do It (The Analogy)

The authors built a mathematical "safety net" with three main parts:

The "Misspecification" Safety Net:
They admit their guess about the network might be slightly off. They create a "fuzziness" parameter (let's call it the "wobble factor"). They ask: "If our guess is off by this much, what is the absolute highest and lowest the vaccine's effectiveness could possibly be?"
- Analogy: Imagine you are guessing the weight of a watermelon. You guess 10 lbs. But you admit you might be off by 2 lbs. So, you calculate that the watermelon is definitely between 8 and 12 lbs. You don't need to know the exact weight to know it's heavy enough to need two hands.
The "Orthogonal" Estimator (The Smart Calculator):
To find these high and low limits, they use a special mathematical trick called "orthogonal estimation."
- Analogy: Imagine you are trying to measure the height of a building, but your tape measure is a bit wobbly and your eyesight is blurry. A normal method would get confused by the wobble and give a wrong answer. This new method is like a "self-correcting" tape measure. It separates the "wobble" (the messy data) from the "height" (the answer) so that even if your guess about the wobble is slightly wrong, your final height calculation stays accurate.
Three Common Scenarios:
They tested this safety net on three common ways people usually guess how networks work:
- The Average: Assuming everyone in your circle matters equally.
- The Threshold: Assuming you only get an effect if more than half your friends are treated.
- The "Second-Degree" Friend: Assuming the effect stops at your direct friends, but maybe it actually reaches your friends' friends.

What They Found (The Results)

The authors ran computer simulations to test their new framework:

It's Safe (Valid): When they made their "wobble factor" big enough to cover the truth, their calculated range always included the real answer. The old methods (which pretend they know the exact rule) often missed the truth completely.
It's Sharp (Tight): The range they gave wasn't just "between 0 and 100%" (which is useless). It was a tight, useful range (e.g., "between 10% and 15%").
It's Fast and Reliable: Their special "self-correcting" calculator worked much better than standard methods, especially when the data was messy or the network was huge.

The Bottom Line

This paper doesn't tell you exactly how a vaccine works in a specific network. Instead, it gives researchers a robust tool to say: "Even if we don't perfectly understand how people influence each other, we can still give you a reliable, safe range for how effective the treatment is."

It turns a "guess and hope" situation into a "calculate the worst and best case" situation, ensuring that decisions based on this data aren't built on a shaky foundation.

Technical Summary: Causal Inference on Networks under Misspecified Exposure Mappings

1. Problem Statement

Estimating treatment effects in network settings is fundamentally challenging because the outcome of a unit depends not only on its own treatment but also on the treatments of connected units (interference). This violates the Stable Unit Treatment Variable Assumption (SUTVA). To manage the exponential complexity of conditioning on all possible treatment assignments in a network of size $N$ , existing methods typically employ an exposure mapping ( $g$ ). This function compresses the high-dimensional treatment assignments of a unit's neighbors into a low-dimensional summary (e.g., the proportion of treated neighbors).

However, these methods rely on the assumption that the exposure mapping is correctly specified based on domain knowledge. In practice, the mechanism of spillover propagation is often only partially understood, leading to misspecification of $g$ . When the assumed mapping $g$ differs from the true underlying mapping $g^*$ , standard estimators for direct and spillover effects become severely biased. Current literature lacks a general framework to bound potential outcomes and treatment effects when the exposure mapping is misspecified, particularly for both experimental and observational data.

2. Methodology

The authors propose a partial identification framework that assesses the robustness of causal estimates under exposure mapping misspecification. Instead of attempting to point-identify the treatment effect under a potentially wrong model, the framework derives sharp upper and lower bounds on the potential outcomes and treatment effects.

Core Components:

Sensitivity Model via Propensity Bounds: The framework models misspecification as a distribution shift between the propensity of the assumed mapping $g$ and the true mapping $g^*$ . It introduces user-specified bounds $b^-(z, x)$ and $b^+(z, x)$ such that the ratio of the true propensity to the assumed propensity is bounded:
$b^-(z, x) \leq \frac{p(g^*(t_N) = z \mid x)}{p(g(t_N) = z \mid x)} \leq b^+(z, x)$
This approach is instantiated for three canonical exposure settings:
1. Weighted Means: Allowing the weights of neighbor treatments to deviate from a uniform mean by a margin $\epsilon$ .
2. Thresholding: Allowing the threshold $c$ used to define exposure (e.g., "more than half treated") to vary within $[c \pm \epsilon]$ .
3. Higher-Order Spillovers: Accounting for interference from neighbors beyond the immediate neighborhood (unobserved confounding), utilizing sensitivity bounds from the unobserved confounding literature.
Sharp Bound Derivation: Using ideas from causal sensitivity analysis, the authors derive closed-form expressions for the sharp upper and lower bounds on the Conditional Average Potential Outcome (CAPO), $\mu^\pm(t, z, x)$ . These bounds are constructed by optimizing the expected outcome over a set of distributions consistent with the observed data and the propensity bounds. The solution involves a cutoff value $Q^\pm$ (a conditional quantile) and conditional moment functions, ensuring the bounds are the narrowest possible intervals containing the true outcome given the misspecification level.
Orthogonal Estimation: To estimate these bounds from data, the authors develop an orthogonal estimator based on statistical learning theory.
- Nuisance Parameters: The estimation requires estimating nuisance functions: the propensity scores ( $\pi_t, \pi_g$ ), the cutoffs ( $Q^\pm$ ), and conditional moment functions ( $\gamma^\pm_u, \gamma^\pm_l$ ).
- Pseudo-Outcome Construction: They construct an orthogonal pseudo-outcome $\phi^\pm_{t,z}$ that is first-order insensitive to errors in the nuisance estimators. This allows the use of flexible machine learning models for the nuisance functions without introducing significant bias into the final bound estimates.
- Two-Stage Procedure: The method employs $K$ -fold cross-fitting. First, nuisance functions are learned on training folds. Second, the pseudo-outcomes are computed on held-out folds and regressed on covariates $X$ to obtain the final bound estimates.

3. Key Contributions

Partial Identification Framework: The paper introduces a novel framework for causal inference on networks that provides sharp, valid bounds on treatment effects under exposure mapping misspecification, addressing a gap where no general framework previously existed.
Theoretical Guarantees: The derived bounds are proven to be sharp (the narrowest possible intervals) and valid (containing the true effect) under the specified sensitivity assumptions.
Orthogonal Estimator: The authors propose a model-agnostic orthogonal estimator that achieves quasi-oracle convergence rates. This means the estimator's error depends only on the product of nuisance estimation errors (second-order), allowing it to remain robust even if the nuisance models are misspecified or converge slowly.
Instantiation for Canonical Settings: The framework is explicitly formulated for three widely used exposure mappings: weighted neighborhood means, threshold-based mappings, and scenarios involving higher-order spillovers.

4. Experimental Results

The authors evaluate the framework on synthetic datasets with varying network sizes ( $N=1000$ and $N=6000$ ) and covariate dimensions.

Validity: Visualizations confirm that when the sensitivity factor (the degree of allowed misspecification) is set sufficiently high (factor $\geq 1$ ), the estimated bounds successfully contain the true potential outcomes and treatment effects. Under insufficient sensitivity assumptions, the bounds fail to cover the truth, as expected.
Convergence: Comparisons between the proposed orthogonal estimator and a standard "plug-in" estimator show that the orthogonal estimator converges faster and maintains valid coverage even at smaller sample sizes. The plug-in estimator suffers from substantial finite-sample bias due to nuisance estimation errors.
Informativeness: The bounds are shown to be informative. For correctly specified sensitivity factors, the average width of the Average Direct Effect (ADE) intervals is approximately 8.71% of the overall outcome range. Furthermore, the bounds correctly identify the direction of the treatment effect (strictly positive or negative), avoiding the trivial case where bounds span zero.

5. Significance and Claims

The paper claims to advance the field of causal inference on networks by providing a robust alternative to standard methods that rely on potentially incorrect exposure mappings. By shifting from point identification to partial identification, the framework allows researchers to assess the sensitivity of their conclusions to model misspecification.

The authors emphasize that their framework is model-agnostic and robust to nuisance estimation errors, making it applicable even when complex, non-parametric models are used to estimate the underlying network propensities and outcomes. The work does not claim to eliminate the need for domain knowledge but rather provides a rigorous mathematical tool to quantify the uncertainty introduced when that knowledge is imperfect. The experiments demonstrate that the resulting bounds remain tight enough to support reliable decision-making, even in the presence of exposure mapping misspecification.