Stochastic Neural Networks for Causal Inference with Missing Confounders

TL;DR

Confounder Imputation with Stochastic Neural Networks (CI-StoNet), which parameterizes the conditional structure of a causal directed acyclic graph using a stochastic neural network and imputes latent confounders via adaptive stochastic-gradient Hamiltonian Monte Carlo and the effect of overlap on estimation accuracy is characterized.

Abstract

Unmeasured confounding is a fundamental obstacle to causal inference from observational data. Latent-variable methods address this challenge by imputing unobserved confounders, yet many lack explicit model-based identification guarantees and are difficult to extend to richer causal structures. We propose Confounder Imputation with Stochastic Neural Networks (CI-StoNet), which parameterizes the conditional structure of a causal directed acyclic graph using a stochastic neural network and imputes latent confounders via adaptive stochastic-gradient Hamiltonian Monte Carlo. Under SUTVA and overlap, and assuming that the structural components of the data-generating process are well approximated by a capacity-controlled sparse deep neural network class, we establish model identification and consistent estimation of the mean potential outcome under a fixed intervention within this class. Although the latent confounder is identifiable only up to reparameterizations that preserve the joint treatment-outcome distribution, the causal estimand is invariant across this observationally equivalent class. We further characterize the effect of overlap on estimation accuracy. Empirical results on simulated and benchmark datasets demonstrate accurate performance, and the framework extends naturally to proxy-variable and multiple-cause settings with overlap diagnostics and bootstrap-based uncertainty quantification.

Figures (7)

• Figure 1: simple confounding
• Figure 2: Diagram of CI-StoNet under simple confounding, where white rectangles represent variables from observed data; light-grey rounded-rectangles represent latent variable to impute; and dark-grey rectangles represent neural network modules to learn respective conditional distributions.
• Figure 3: Other examples of causal structures: (a) existence of colliders, represented by $C$ ; (b) existence of mediators, represented by $M$.
• Figure 4: (a) Causal DAG: without dependence on the proxy; (b) Diagram of CI-StoNet under the proxy setting: white rectangles represent variables from observed data; light-grey rounded-rectangles represent hidden neurons; dark-grey rectangles represent network modules to learn respective conditional distributions.
• Figure S1: CI-StoNet results for the simulation study. Points denote the true marginal effects and error bars represent one standard error of the marginal effect estimator.

Theorems & Definitions (26)

• Theorem 1
• Remark 1
• Theorem 2
• Lemma A1
• proof
• Remark A1
• Lemma A2
• Lemma A3
• Lemma A4
• proof