Coupling Generative Modeling and an Autoencoder with the Causal Bridge

We consider inferring the causal effect of a treatment (intervention) on an outcome of interest in situations where there is potentially an unobserved confounder influencing both the treatment and the outcome. This is achievable by assuming access to two separate sets of control (proxy) measurements associated with treatment and outcomes, which are used to estimate treatment effects through a function termed the causal bridge (CB). We present a new theoretical perspective, associated assumptions for when estimating treatment effects with the CB is feasible, and a bound on the average error of the treatment effect when the CB assumptions are violated. From this new perspective, we then demonstrate how coupling the CB with an autoencoder architecture allows for the sharing of statistical strength between observed quantities (proxies, treatment, and outcomes), thus improving the quality of the CB estimates. Experiments on synthetic and real-world data demonstrate the effectiveness of the proposed approach in relation to the state-of-the-art methodology for proxy measurements. A promising approach to address this challenge is the use of proxy variables [8, 9, 10], also known as negative control variables. These are variables that are affected by the unobserved confounder, but do not necessarily directly influence the treatment or outcome. Consequently, leveraging information from these proxies, we can gain insight into the underlying causal mechanisms and potentially mitigate the bias caused by unobserved confounders. Recent work has introduced the concept of a causal bridge function, which uses two sets of proxy variables, one related to the treatment and the other to the outcome, to estimate causal effects [9, 11, 12] . This approach has shown promising results, but there is still room for improvement in terms of both theoretical understanding and practical implementation. This paper builds on the causal bridge framework and makes several key advances. We provide a refined theoretical analysis, clarifying the assumptions and conditions under which the causal bridge function yields accurate causal effect estimates. Furthermore, we introduce a novel learning approach that leverages the power of generative models to enhance the estimation of the causal bridge. Our approach enables the sharing of statistical strength between observed variables, leading to more robust and accurate causal inference. Finally, we extend the causal bridge framework to handle survival outcomes, a common type of data in biomedical applications. Preprint. Under review.