Doubly Robust Proxy Causal Learning with Neural Mean Embeddings

TL;DR

This paper introduces a neural doubly robust framework for proxy causal learning that effectively combines outcome and treatment bridges using neural mean embeddings, improving causal inference in observational studies.

Contribution

It develops a novel neural mean-embedding estimator for treatment bridges and integrates it with outcome bridges for doubly robust causal effect estimation.

Findings

Outperforms existing baselines and single-bridge neural estimators on synthetic and image benchmarks.

Provides a consistent algorithm with controlled doubly robust error.

Estimates full response-curve functions for continuous and structured treatments.

Abstract

Unobserved confounding prevents standard covariate adjustment from identifying causal response functions in observational studies. Proxy causal learning addresses this problem through bridge equations involving treatment- and outcome-inducing proxies, avoiding direct recovery of the latent confounder. Existing doubly robust proxy estimators combine outcome and treatment bridges, but typically rely on fixed kernels, sieves, or low-dimensional semiparametric models; existing neural proxy methods are more flexible, but are largely single-bridge estimators. We develop a neural doubly robust framework for proxy causal learning with continuous and structured treatments. Our method introduces a neural mean-embedding estimator for the treatment bridge, combines it with a neural outcome bridge, and estimates the doubly robust correction through a final regression stage. The framework covers population, heterogeneous, and conditional dose-response functions, yielding full response-curve estimators rather than binary-treatment effects. The algorithms use two stages for each bridge and history-aware updates of the final linear layers to stabilize stochastic multi-stage training. We prove consistency of the algorithms showing that the doubly robust error is controlled by the final averaging and regression errors together with the smaller of the outcome- and treatment-side weak-norm bridge errors. Across synthetic and image-valued benchmarks, the proposed estimators outperform existing baselines and single-bridge neural estimators, showing the benefit of combining learned outcome and treatment bridges in a doubly robust construction. Our implementation is available at https://github.com/BariscanBozkurt/DRPCL-Neural-Mean-Embedding.