Supervised learning datasets often have privileged information, in the form of features which are available at training time but are not available at test time e.g. the ID of the annotator that provided the label. We argue that privileged information is useful for explaining away label noise, thereby reducing the harmful impact of noisy labels. We develop a simple and efficient method for supervised learning with neural networks: it transfers via weight sharing the knowledge learned with privileged information and approximately marginalizes over privileged information at test time. Our method, TRAM (TRansfer and Marginalize), has minimal training time overhead and has the same test-time cost as not using privileged information. TRAM performs strongly on CIFAR-10H, ImageNet and Civil Comments benchmarks.