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Bayesian Estimation of Causal Effects Using Proxies of a Latent Interference Network

This paper proposes a Bayesian inference framework utilizing a Block Gibbs sampler with Locally Informed Proposals to estimate causal effects in the presence of network interference when only proxy measurements of the true interference network are available.

Original authors: Bar Weinstein, Daniel Nevo

Published 2026-02-04

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CC BY 4.0

Original authors: Bar Weinstein, Daniel Nevo

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 how a rumor spreads through a group of friends. To do this, you need to know exactly who talks to whom. This "who talks to whom" map is the interference network. If you know the map perfectly, you can predict how a rumor (or a vaccine, or a new policy) will ripple through the group.

However, in the real world, researchers rarely have the perfect map. They usually have proxies—imperfect, blurry, or incomplete copies of the truth. Maybe they asked people who their friends are (and people forgot some names), or maybe they looked at who "liked" each other's posts on Facebook, even though the real influence happens in person at the coffee shop.

This paper, by Bar Weinstein and Daniel Nevo, tackles the problem of estimating the true effects of a policy when you only have these blurry, imperfect maps.

The Core Problem: The "Blurred Photo" Dilemma

Think of the true network of influence as a high-definition photo of a crowd. The data researchers collect is like a series of photos taken through a foggy window, or perhaps photos taken from different angles that don't quite match.

Traditionally, researchers would just pick one of these foggy photos, assume it's the real thing, and run their calculations. The authors argue this is dangerous. If you treat a blurry photo as the truth, your predictions about how a rumor spreads will be wrong.

The Solution: A "Detective" Approach

Instead of picking one blurry photo and pretending it's perfect, the authors propose a Bayesian Detective Framework.

Imagine you are a detective trying to reconstruct the crime scene (the true network) using three different types of clues:

The Foggy Photos (Proxy Networks): The imperfect maps of who knows whom.
The Witness Statements (Outcomes): What actually happened to the people (e.g., did they get sick? did they buy the product?).
The Suspect List (Treatments): Who was given the "treatment" (like a vaccine or a new rule).

The authors' method doesn't just look at the photos. It uses all three clues together to guess what the high-definition photo must have looked like. If the witness statements say "Person A got sick immediately after Person B," but the foggy photo says they don't know each other, the detective (the algorithm) realizes the photo is wrong and updates its guess of the true network.

The Engine: The "Smart Search" (Block Gibbs Sampler)

Reconstructing the true network is like trying to solve a massive puzzle where the pieces keep changing shape. There are billions of possible ways the network could be arranged. A computer trying to guess the right arrangement by flipping one piece at a time randomly would take forever.

To solve this, the authors built a Block Gibbs Sampler with Locally Informed Proposals.

The Analogy: Imagine you are in a dark room trying to find the exit. A random walker would bump into walls randomly. A "Locally Informed" walker, however, uses a flashlight to feel the air currents and the texture of the walls to guess which direction leads to the exit right now.
How it works: The algorithm doesn't just guess randomly. It looks at the current state of the network and uses the "clues" (outcomes and treatments) to intelligently suggest which connections are likely real and which are fake. It flips the "edges" (connections) of the network in a way that is most likely to be correct, based on all the data it has.

What They Found

The authors tested their method with computer simulations (creating fake data where they knew the "true" network) and real-world data (social networks from a university).

The Result: Their "Detective" method was much better at finding the true network and calculating the true effects of policies than the old methods.
The "Two-Stage" Trap: They showed that a common shortcut—first guessing the network, then using that guess to calculate effects—often leads to mistakes and makes researchers overconfident in their wrong answers. Their method avoids this by guessing the network and the effects at the same time.
Robustness: Even when the "foggy photos" were very bad (very noisy), their method could still recover the truth by listening closely to the "witness statements" (outcomes).

In a Nutshell

This paper gives researchers a new, smarter way to do causal analysis when they don't have perfect data. Instead of ignoring the messiness of real-world data or making dangerous assumptions, they use a sophisticated statistical engine that treats the true network as a mystery to be solved using every scrap of evidence available. It's like upgrading from guessing the shape of a cloud to using a 3D scanner that reconstructs the object from multiple blurry angles.

Technical Summary: Bayesian Estimation of Causal Effects Using Proxies of a Latent Interference Network

Problem Statement

In causal inference, network interference occurs when the treatment assigned to one unit affects the outcomes of others. Traditional approaches typically assume that the observed network correctly specifies the interference structure. However, in practice, researchers often lack access to the true interference network ( $A^*$ ). Instead, they possess only proxy measurements ( $A$ ) due to data collection limitations (e.g., self-reported errors, recall bias) or because the observed social ties do not correspond to the actual pathways of interference (e.g., online connections vs. physical contact).

Treating a proxy network as the true interference structure can lead to model misspecification and biased causal effect estimates. Existing solutions often rely on specific structural assumptions (e.g., linear-in-means models with specific error mechanisms) or two-stage approaches that first infer the network and then treat it as fixed. The latter approach fails to propagate uncertainty from the network inference stage to the causal estimates, leading to underestimation of uncertainty. Furthermore, the true interference network is often latent, creating a high-dimensional discrete space that makes standard Bayesian inference computationally intractable.

Methodology

Structural Causal Models (SCMs)

The authors propose a general Structural Causal Model framework where the true interference network $A^*$ is treated as a latent variable generated from a random network model. The framework accommodates diverse structural configurations:

Causal Proxies: The observed networks $A$ are direct descendants of $A^*$ (e.g., noisy measurements).
Non-Causal Proxies: Both $A^*$ and $A$ depend on shared latent variables $V$ (e.g., multilayer networks sharing latent positions), with no direct causal link between them.
Treatment Assignment: The framework models treatment assignment ( $Z$ ) as potentially dependent on either the latent network $A^*$ (observational settings) or the observed proxies $A$ (randomized trials with cluster randomization based on proxies).

The outcome model ( $Y$ ) incorporates interference through summarizing functions (exposure mappings) that aggregate treatments and covariates of neighbors based on the true network $A^*$ . The causal estimands are defined as interventions on population-level treatments (static, dynamic, or stochastic policies), distinct from interventions on exposure levels.

Identifiability

The paper investigates the identifiability of structural parameters in simplified settings. It demonstrates that while the latent nature of $A^*$ poses challenges, the structural parameters are recoverable provided there is sufficient information from the joint distribution of outcomes, treatments, and proxy networks. The analysis shows that each data module (outcomes, treatments, proxies) contributes to identifying the parameters, often requiring fewer independent proxy networks than would be needed if only the proxies were used.

Bayesian Inference and Sampling Algorithm

To estimate the causal effects, the authors develop a Bayesian inference framework. The posterior distribution involves a mixed space of continuous parameters ( $\Theta$ ) and the discrete latent network ( $A^*$ ). Direct marginalization of $A^*$ is computationally infeasible due to the super-exponential growth of the state space ( $O(2^{N^2})$ ).

To address this, the authors propose a Block Gibbs sampler with Locally Informed Proposals (LIP):

Block Gibbs Structure: The algorithm alternates between updating continuous parameters (using standard MCMC techniques like NUTS) and updating the discrete latent network $A^*$ .
Locally Informed Proposals (LIP): Standard random walk proposals for network updates suffer from poor mixing in high-dimensional discrete spaces. The authors employ LIP, which biases edge flips toward states with higher posterior probability.
Gradient-Based Approximation: Exact LIP requires evaluating the posterior for every possible edge flip, which is $O(N^2)$ per iteration. To make this feasible, the authors approximate the log-posterior differences using gradients (via a first-order Taylor series). This reduces the computational cost to a single gradient evaluation per iteration, leveraging automatic differentiation.
Initialization: A multi-stage initialization procedure is used to ensure the Markov chain starts in a high-probability region, utilizing cut-posterior distributions to estimate network and proxy parameters before refining the network state with full model feedback.

Causal Effect Estimation

Causal estimands are estimated by averaging over the posterior samples of the latent network and parameters. For a static policy, the expected outcome is approximated by averaging the conditional expectations over the posterior samples of $A^*$ and $\Theta$ . This approach naturally propagates uncertainty from the network inference to the final causal estimates.

Key Contributions

Flexible Structural Causal Models: The paper introduces a general SCM framework that treats the interference network as a latent variable, accommodating both causal and non-causal proxy mechanisms and various treatment assignment dependencies.
Joint Estimation: Unlike heuristic two-stage approaches, the proposed framework jointly estimates the latent network and causal parameters. This ensures that uncertainty in the network structure is correctly propagated to the causal effect estimates.
Efficient Inference for Discrete Latent Networks: The authors develop a Block Gibbs sampler equipped with Locally Informed Proposals and gradient-based approximations. This enables efficient exploration of the high-dimensional discrete posterior space, making Bayesian inference feasible for network sizes where brute-force marginalization is impossible. The paper also provides a formal analysis of the approximation error, showing it remains independent of network size under certain conditions.

Results

The authors evaluate the method using fully-synthetic and semi-synthetic data:

Fully-Synthetic Data: Simulations with $N=500$ units demonstrate that the proposed Block Gibbs (BG) algorithm consistently achieves lower Mean Absolute Percentage Error (MAPE) in estimating causal effects compared to baselines, including naive approaches (treating proxies as truth), continuous relaxation methods, and two-stage approaches. Crucially, the BG method achieves empirical coverage rates for credible intervals close to the nominal level, whereas two-stage methods and aggregation-based baselines tend to underestimate uncertainty.
Semi-Synthetic Data: Using a real-world multilayer network dataset (Aarhus University employees), the method was tested by designating one layer as the latent network and others as proxies. The BG method again outperformed two-stage methods and simple aggregation strategies (union/intersection of layers) in terms of estimation accuracy and uncertainty quantification.

Significance and Claims

The paper claims to advance the literature on causal inference with network interference by providing a unified framework for scenarios where the interference network is latent and only proxy measurements are available. The significance lies in:

Addressing Misspecification: It explicitly models the discrepancy between observed proxies and the true interference structure, mitigating biases caused by network misspecification.
Uncertainty Quantification: By jointly estimating the network and causal effects, the framework avoids the underestimation of uncertainty inherent in two-stage approaches.
Computational Feasibility: The development of the gradient-based LIP within a Block Gibbs sampler makes it computationally feasible to perform Bayesian inference on latent networks of practical sizes, a task previously hindered by the discrete nature of the problem.

The authors note that their approach relies on accurate model specification, which can be assessed via posterior predictive checks. They also acknowledge that while their framework handles latent networks, identifying effects of interventions on the network structure itself may still require additional assumptions due to latent confounding.