Causal Effect Estimation using identifiable Variational AutoEncoder with Latent Confounders and Post-Treatment Variables

TL;DR

This work tackles unbiased causal effect estimation from observational data in the presence of latent confounders and post-treatment variables. It introduces CPTiVAE, which jointly learns latent confounders via VAE from proxy $ ext{X}_ ext{C}$ and latent post-treatment variables via identifiable VAE from proxy $ ext{X}_ ext{M}$, incorporating identifiability guarantees for the latent post-treatment representation. The authors prove a theorem establishing $ ext{M}$-identifiability under mild conditions and demonstrate through synthetic, semi-synthetic (IHDP-based), and real-world (Adult) experiments that CPTiVAE outperforms state-of-the-art baselines in estimating $ATE$ and $CATE$, while maintaining robustness to hyperparameters and latent dimension choices. The approach offers a principled framework to adjust for both confounding and post-treatment biases, with practical implications for causal analysis in domains where post-treatment pathways are informative but unobserved. The work also provides public code, underscoring its potential for broad adoption in causal inference tasks that leverage proxy information.

Abstract

Estimating causal effects from observational data is challenging, especially in the presence of latent confounders. Much work has been done on addressing this challenge, but most of the existing research ignores the bias introduced by the post-treatment variables. In this paper, we propose a novel method of joint Variational AutoEncoder (VAE) and identifiable Variational AutoEncoder (iVAE) for learning the representations of latent confounders and latent post-treatment variables from their proxy variables, termed CPTiVAE, to achieve unbiased causal effect estimation from observational data. We further prove the identifiability in terms of the representation of latent post-treatment variables. Extensive experiments on synthetic and semi-synthetic datasets demonstrate that the CPTiVAE outperforms the state-of-the-art methods in the presence of latent confounders and post-treatment variables. We further apply CPTiVAE to a real-world dataset to show its potential application.

Figures (5)

• Figure 1: The causal graph assumed by CEVAE, TEDVAE, and CPTiVAE. In all figures, $T$ is the treatment, $Y$ is the outcome, and $\mathbf{C}$ are the latent confounders that affect both treatment and outcome. (a) The causal graph of CEVAE. $\mathbf{X}$ is the proxy for $\mathbf{C}$. (b) The causal graph of TEDVAE, $\mathbf{X}$ is the observed variables which may contain non-confounders and noisy proxy variables, $\mathbf{L}$ are factors that affect only the treatment, $\mathbf{F}$ are factors that affect only the outcome. (c) The causal graph for the proposed CPTiVAE. $\mathbf{M}$ are the latent post-treatment variables which are affected by the treatment variable $T$, $\mathbf{X_{C}}$ and $\mathbf{X_{M}}$ are proxies for $\mathbf{C}$ and $\mathbf{M}$, respectively.
• Figure 2: The results of the dimensionality study. "True dimensions" refer to the dimensions of $\mathbf{C}$ and $\mathbf{M}$ in the data, and "Setting dimensions" correspond to the parameters of $D_{\mathbf{C}}$ and $D_{\mathbf{M}}$ in the CPTiVAE algorithm.
• Figure 3: The performance for estimating ATE and CATE on the semi-synthetic dataset.
• Figure 4: The causal network for the Adult dataset: the green path represents the direct path, and the blue paths represent the indirect paths passing through marital status ijcai2017p549.
• Figure 5: Simplified DAG for Adult dataset.

Theorems & Definitions (8)

• Definition 1: Markov property pearl_2009
• Definition 2: Faithfulness spirtes2000causation
• Definition 3: $d$-separation pearl_2009
• Definition 4: Back-door criterion pearl_2009
• Definition 5: Identifiability classes
• Definition 6
• Theorem 1
• proof