Gaussian Processes (GPs) are effective Bayesian predictors. We here show for the first time that instance labels of a GP classifier can be inferred in the multiple instance learning (MIL) setting using variational Bayes. We achieve this via a new construction of the bag likelihood that assumes a large value if the instance predictions obey the MIL constraints and a small value otherwise. This construction lets us derive the update rules for the variational parameters analytically, assuring both scalable learning and fast convergence. We observe this model to improve the state of the art in instance label prediction from bag-level supervision in the 20 Newsgroups benchmark, as well as in Barrett's cancer tumor localization from histopathology tissue microarray images. Furthermore, we introduce a novel pipeline for weakly supervised object detection naturally complemented with our model, which improves the state of the art on the PASCAL VOC 2007 and 2012 data sets. Last but not least, the performance of our model can be further boosted up using mixed supervision: a combination of weak (bag) and strong (instance) labels.