  What is the best way to exploit extra data -- be it unlabeled data from the
same task, or labeled data from a related task -- to learn a given task? This
paper formalizes the question using the theory of reference priors. Reference
priors are objective, uninformative Bayesian priors that maximize the mutual
information between the task and the weights of the model. Such priors enable
the task to maximally affect the Bayesian posterior, e.g., reference priors
depend upon the number of samples available for learning the task and for very
small sample sizes, the prior puts more probability mass on low-complexity
models in the hypothesis space. This paper presents the first demonstration of
reference priors for medium-scale deep networks and image-based data. We
develop generalizations of reference priors and demonstrate applications to two
problems. First, by using unlabeled data to compute the reference prior, we
develop new Bayesian semi-supervised learning methods that remain effective
even with very few samples per class. Second, by using labeled data from the
source task to compute the reference prior, we develop a new pretraining method
for transfer learning that allows data from the target task to maximally affect
the Bayesian posterior. Empirical validation of these methods is conducted on
image classification datasets. Code is available at
https://github.com/grasp-lyrl/deep_reference_priors.
