Modern machine learning demands a large amount of training data. Weak supervision is a promising approach to meet this demand. It aggregates multiple labeling functions (LFs)—noisy, user-provided labeling heuristics---to rapidly and cheaply curate probabilistic labels for large-scale unlabeled data. However, standard assumptions in weak supervision---such as user-specified class balance, similar accuracy of an LF in classifying different classes, and full knowledge of LF dependency at inference time---might be undesirable in practice. In response, we present Firebolt, a new weak supervision framework that seeks to operate under weaker assumptions. In particular, Firebolt learns the class balance and class-specific accuracy of LFs jointly from unlabeled data. It carries out inference in an efficient and interpretable manner. We analyze the parameter estimation error of Firebolt and characterize its impact on downstream model performance. Furthermore, we show that on five publicly available datasets, Firebolt outperforms a state-of-the-art weak supervision method by up to 5.8 points in AUC. We also provide a case study in the production setting of a tech company, where a Firebolt-supervised model outperforms the existing weakly-supervised production model by 1.3 points in AUC and speedup label model training and inference from one hour to three minutes.