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

Bar Weinstein, Daniel Nevo

TL;DR

This paper addresses causal inference under network interference when only proxy measurements of a latent interference network are available. It proposes a flexible structural causal-modeling framework that treats the true interference network as latent and derives a Bayesian approach that jointly imputes the latent network and estimates causal effects, propagating uncertainty through to policy evaluations. A key methodological contribution is a Block Gibbs sampler with Locally Informed Proposals for efficient inference on the large discrete network space, aided by gradient-based approximations. Empirical results on fully- and semi-synthetic data demonstrate accurate recovery of causal effects and well-calibrated uncertainty, outperforming naive and two-stage methods, with practical utility for policy evaluation under imperfect network information.

Abstract

Network interference occurs when treatments assigned to some units affect the outcomes of others. Traditional approaches often assume that the observed network correctly specifies the interference structure. However, in practice, researchers frequently only have access to proxy measurements of the interference network due to limitations in data collection or potential mismatches between measured networks and actual interference pathways. In this paper, we introduce a framework for estimating causal effects when only proxy networks are available. Our approach leverages a structural causal model that accommodates diverse proxy types, including noisy measurements, multiple data sources, and multilayer networks, and defines causal effects as interventions on population-level treatments. The latent nature of the true interference network poses significant challenges. To overcome them, we develop a Bayesian inference framework. We propose a Block Gibbs sampler with Locally Informed Proposals to update the latent network, thereby efficiently exploring the high-dimensional posterior space composed of both discrete and continuous parameters. The latent network updates are driven by information from the proxy networks, treatments, and outcomes. We illustrate the performance of our method through numerical experiments, demonstrating its accuracy in recovering causal effects even when only proxies of the interference network are available.

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

TL;DR

This paper addresses causal inference under network interference when only proxy measurements of a latent interference network are available. It proposes a flexible structural causal-modeling framework that treats the true interference network as latent and derives a Bayesian approach that jointly imputes the latent network and estimates causal effects, propagating uncertainty through to policy evaluations. A key methodological contribution is a Block Gibbs sampler with Locally Informed Proposals for efficient inference on the large discrete network space, aided by gradient-based approximations. Empirical results on fully- and semi-synthetic data demonstrate accurate recovery of causal effects and well-calibrated uncertainty, outperforming naive and two-stage methods, with practical utility for policy evaluation under imperfect network information.

Abstract

Network interference occurs when treatments assigned to some units affect the outcomes of others. Traditional approaches often assume that the observed network correctly specifies the interference structure. However, in practice, researchers frequently only have access to proxy measurements of the interference network due to limitations in data collection or potential mismatches between measured networks and actual interference pathways. In this paper, we introduce a framework for estimating causal effects when only proxy networks are available. Our approach leverages a structural causal model that accommodates diverse proxy types, including noisy measurements, multiple data sources, and multilayer networks, and defines causal effects as interventions on population-level treatments. The latent nature of the true interference network poses significant challenges. To overcome them, we develop a Bayesian inference framework. We propose a Block Gibbs sampler with Locally Informed Proposals to update the latent network, thereby efficiently exploring the high-dimensional posterior space composed of both discrete and continuous parameters. The latent network updates are driven by information from the proxy networks, treatments, and outcomes. We illustrate the performance of our method through numerical experiments, demonstrating its accuracy in recovering causal effects even when only proxies of the interference network are available.
Paper Structure (45 sections, 1 theorem, 111 equations, 12 figures, 3 tables, 2 algorithms)

This paper contains 45 sections, 1 theorem, 111 equations, 12 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Our Contribution
  4. Flexible Structural Causal Models.
  5. Joint Estimation.
  6. Inference for Discrete Latent Networks.
  7. Settings and Causal Models
  8. Setup and Notations
  9. Structural Causal Models
  10. Causal Estimands
  11. Probabilistic Models
  12. Proxy Measurements and Network Models
  13. Types of Proxy Networks
  14. True and Proxy Network Models
  15. Identifiability of Structural Parameters
  16. ...and 30 more sections

Key Result

Theorem 2.2

Let $\mathscr{I}_{obs, kk}(\boldsymbol{\Theta}_0) = \mathscr{I}_{comp, kk}(\boldsymbol{\Theta}_0) - \mathscr{I}_{miss, kk}(\boldsymbol{\Theta}_0)$ be the observed data FIM in module $k$ at point $\boldsymbol{\Theta}_0$. The parameters $\boldsymbol{\Theta}$ are locally identifiable at point $\boldsym where $\lambda_{\min}$ denotes the minimum eigenvalue and $\lVert \cdot \rVert_2$ is the spectral n

Figures (12)

  • Figure 1: DAGs representing the assumed structural models for four different settings, corresponding to causal or non-causal proxies and treatment assignment based on the latent network $\boldsymbol{A}^\ast$ or the proxies $\mathcal{A}$: (a) Causal proxies with latent assignment. (b) Causal proxies with proxy assignment. (c) Non-causal proxies with latent assignment. (d) Non-causal proxies with proxy assignment. The dashed arrows are optional.
  • Figure 2: Mean ($\pm \; \text{SD}$) of MAPE values across $300$ simulation iterations for dynamic and TTE estimands. Proxy networks $\mathcal{A}$ are weaker as $\gamma_2$ (x-axis) increases. Smaller values indicate better performance.
  • Figure 3: Heatmap of true $\boldsymbol{A}^\ast$, observed proxy $\boldsymbol{A}_1$, and posterior probabilities under one iteration with $\gamma_2=3$. In the True and Observed heatmaps, entries are binary. The posterior probabilities are the proportion of times an edge existed in the posterior network samples using a Block Gibbs sampler with a single proxy network. Nodes were first rearranged by hierarchical clustering for clearer visualization.
  • Figure 4.1: Gradient approximations $\widetilde{\Delta}\left(ij,t \mid \cdot \right)$ versus the differences $\Delta\left(ij,t \mid \cdot \right)$. Values are on the log-softmax scale. The approximation is accurate up to a scale.
  • Figure 4.2: Degree distribution of $\boldsymbol{A}^\ast$ in one iteration.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Example 1: Random noise
  • Example 2: Censoring
  • Example 3: Cross-cluster contamination
  • Example 4: Repeated noisy measurements
  • Example 5: Multilayer networks
  • Remark 1: From Structural Parameters to Causal Estimands
  • Definition 2.1: Global Identification via Moments
  • Theorem 2.2: Local Identifiability