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Abstract Semi-supervised (SS) inference has received much attention in recent years. Apart from a moderate-sized labeled data, $$\mathcal L$$, the SS setting is characterized by an additional, much larger sized, unlabeled data, $$\mathcal U$$. The setting of $$|\mathcal U\ |\gg |\mathcal L\ |$$, makes SS inference unique and different from the standard missing data problems, owing to natural violation of the so-called ‘positivity’ or ‘overlap’ assumption. However, most of the SS literature implicitly assumes $$\mathcal L$$ and $$\mathcal U$$ to be equally distributed, i.e., no selection bias in the labeling. Inferential challenges in missing at random type labeling allowing for selection bias, are inevitably exacerbated by the decaying nature of the propensity score (PS). We address this gap for a prototype problem, the estimation of the response’s mean. We propose a double robust SS mean estimator and give a complete characterization of its asymptotic properties. The proposed estimator is consistent as long as either the outcome or the PS model is correctly specified. When both models are correctly specified, we provide inference results with a non-standard consistency rate that depends on the smaller size $$|\mathcal L\ |$$. The results are also extended to causal inference with imbalanced treatment groups. Further, we provide several novel choices of models and estimators of the decaying PS, including a novel offset logistic model and a stratified labeling model. We present their properties under both high- and low-dimensional settings. These may be of independent interest. Lastly, we present extensive simulations and also a real data application.
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This content will become publicly available on April 21, 2026
The Decaying Missing-at-Random Framework: Model Doubly Robust Causal Inference with Partially Labeled Data
In modern large-scale observational studies, data collection constraints often result in partially labeled datasets, posing challenges for reliable causal inference, especially due to potential labeling bias and relatively small size of the labeled data. This paper introduces a decaying missing-at-random (decaying MAR) framework and associated approaches for doubly robust causal inference on treatment effects in such semi-supervised (SS) settings. This simultaneously addresses selection bias in the labeling mechanism and the extreme imbalance between labeled and unlabeled groups, bridging the gap between the standard SS and missing data literatures, while throughout allowing for confounded treatment assignment and high-dimensional confounders under appropriate sparsity conditions. To ensure robust causal conclusions, we propose a bias-reduced SS (BRSS) estimator for the average treatment effect, a type of 'model doubly robust' estimator appropriate for such settings, establishing asymptotic normality at the appropriate rate under decaying labeling propensity scores, provided that at least one nuisance model is correctly specified. Our approach also relaxes sparsity conditions beyond those required in existing methods, including standard supervised approaches. Recognizing the asymmetry between labeling and treatment mechanisms, we further introduce a de-coupled BRSS (DC-BRSS) estimator, which integrates inverse probability weighting (IPW) with bias-reducing techniques in nuisance estimation. This refinement further weakens model specification and sparsity requirements. Numerical experiments confirm the effectiveness and adaptability of our estimators in addressing labeling bias and model misspecification.
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- Award ID(s):
- 2113768
- PAR ID:
- 10650010
- Publisher / Repository:
- arXiv.org
- Date Published:
- Journal Name:
- arXiv.org
- Edition / Version:
- arXiv preprint arXiv:2305.12789v3
- ISSN:
- 2331-8422
- Page Range / eLocation ID:
- 1-72
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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