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In real-world scenarios, data collection limitations often result inpartially labeled datasets, leading to difficulties in drawing reliable causalinferences. Traditional approaches in the semi-supervised (SS) and missing dataliterature may not adequately handle these complexities, leading to biasedestimates. To address these challenges, our paper introduces a novel decayingmissing-at-random (decaying MAR) framework. This framework tackles missingoutcomes in high-dimensional settings and accounts for selection bias arisingfrom the dependence of labeling probability on covariates. Notably, we relaxthe need for a positivity condition, commonly required in the missing dataliterature, and allow uniform decay of labeling propensity scores with samplesize, accommodating faster growth of unlabeled data. Our decaying MAR frameworkenables easy rate double-robust (DR) estimation of average treatment effects,succeeding where other methods fail, even with correctly specified nuisancemodels. Additionally, it facilitates asymptotic normality under modelmisspecification. To achieve this, we propose adaptive new targetedbias-reducing nuisance estimators and asymmetric cross-fitting, along with anovel semi-parametric approach that fully leverages large volumes of unlabeleddata. Our approach requires weak sparsity conditions. Numerical results confirmour estimators' efficacy and versatility, addressing selection bias and modelmisspecification.

The Decaying Missing-at-Random Framework: Doubly Robust Causal Inference with Partially Labeled Data