  Labeling data for modern machine learning is expensive and time-consuming.
Latent variable models can be used to infer labels from weaker,
easier-to-acquire sources operating on unlabeled data. Such models can also be
trained using labeled data, presenting a key question: should a user invest in
few labeled or many unlabeled points? We answer this via a framework centered
on model misspecification in method-of-moments latent variable estimation. Our
core result is a bias-variance decomposition of the generalization error, which
shows that the unlabeled-only approach incurs additional bias under
misspecification. We then introduce a correction that provably removes this
bias in certain cases. We apply our decomposition framework to three scenarios
-- well-specified, misspecified, and corrected models -- to 1) choose between
labeled and unlabeled data and 2) learn from their combination. We observe
theoretically and with synthetic experiments that for well-specified models,
labeled points are worth a constant factor more than unlabeled points. With
misspecification, however, their relative value is higher due to the additional
bias but can be reduced with correction. We also apply our approach to study
real-world weak supervision techniques for dataset construction.
