  We study the optimal rates of convergence for estimating a prior distribution
over a VC class from a sequence of independent data sets respectively labeled
by independent target functions sampled from the prior. We specifically derive
upper and lower bounds on the optimal rates under a smoothness condition on the
correct prior, with the number of samples per data set equal the VC dimension.
These results have implications for the improvements achievable via transfer
learning. We additionally extend this setting to real-valued function, where we
establish consistency of an estimator for the prior, and discuss an additional
application to a preference elicitation problem in algorithmic economics.
