  In biomedical studies, it is often desirable to characterize the interactive
mode of multiple disease outcomes beyond their marginal risk. Ising model is
one of the most popular choices serving for this purpose. Nevertheless,
learning efficiency of Ising models can be impeded by the scarcity of accurate
disease labels, which is a prominent problem in contemporary studies driven by
electronic health records (EHR). Semi-supervised learning (SSL) leverages the
large unlabeled sample with auxiliary EHR features to assist the learning with
labeled data only and is a potential solution to this issue. In this paper, we
develop a novel SSL method for efficient inference of Ising model. Our method
first models the outcomes against the auxiliary features, then uses it to
project the score function of the supervised estimator onto the EHR features,
and incorporates the unlabeled sample to augment the supervised estimator for
variance reduction without introducing bias. For the key step of conditional
modeling, we propose strategies that can effectively leverage the auxiliary EHR
information while maintaining moderate model complexity. In addition, we
introduce approaches including intrinsic efficient updates and ensemble, to
overcome the potential misspecification of the conditional model that may cause
efficiency loss. Our method is justified by asymptotic theory and shown to
outperform existing SSL methods through simulation studies. We also illustrate
its utility in a real example about several key phenotypes related to frequent
ICU admission on MIMIC-III data set.
