Sepsis is the leading cause of in-hospital mortality in the USA. Early sepsis onset prediction and diagnosis could significantly improve the survival of sepsis patients. Existing predictive models are usually trained on high-quality data with few missing information, while missing values widely exist in real-world clinical scenarios (especially in the first hours of admissions to the hospital), which causes a significant decrease in accuracy and an increase in uncertainty for the predictive models. The common method to handle missing values is imputation, which replaces the unavailable variables with estimates from the observed data. The uncertainty of imputation results can be propagated to the sepsis prediction outputs, which have not been studied in existing works on either sepsis prediction or uncertainty quantification. In this study, we first define such propagated uncertainty as the variance of prediction output and then introduce uncertainty propagation methods to quantify the propagated uncertainty. Moreover, for the potential high-risk patients with low confidence due to limited observations, we propose a robust active sensing algorithm to increase confidence by actively recommending clinicians to observe the most informative variables. We validate the proposed models in both publicly available data (i.e., MIMIC-III and AmsterdamUMCdb) and proprietary data in The Ohio State University Wexner Medical Center (OSUWMC). The experimental results show that the propagated uncertainty is dominant at the beginning of admissions to hospitals and the proposed algorithm outperforms state-of-the-art active sensing methods. Finally, we implement a SepsisLab system for early sepsis prediction and active sensing based on our pre-trained models. Clinicians and potential sepsis patients can benefit from the system in early prediction and diagnosis of sepsis.