Coarsening at random for multiple instance learning: identifiability conditions for instance-level inference

Multiple instance learning (MIL) trains on bags of instances labelled only at the bag level: a bag is positive when at least one of its instances is positive, and the instance labels are latent. We show that MIL is an instance of the masked-data identifiability problem from reliability statistics: instances are components, bags are candidate sets, the logical-OR bag rule is series aggregation, and singleton bags (instance-level labels) are singleton candidate sets that restore identifiability. We establish a rank condition for identifiability of instance-type positivity and its continuous-feature extension, a bag-prevalence consistency theorem (an IRLS-weighted moment-matching identity at the interior MLE), and a first-order bias bound under aggregation-rule misspecification. The standard-versus-collective MIL taxonomy is recovered as a partition of aggregation rules by support-condition preservation, and a confound between intrinsic instance positivity and the noisy-OR firing rate is identified as the discrete-label analogue of the spike-in capture-efficiency gap in scRNA-seq. Simulation and an application to the MUSK1 and MUSK2 benchmarks confirm the theory.