  We consider a new learning model in which a joint distribution over vector
pairs $(x,y)$ is determined by an unknown function $c(x)$ that maps input
vectors $x$ not to individual outputs, but to entire {\em distributions\/} over
output vectors $y$. Our main results take the form of rather general reductions
from our model to algorithms for PAC learning the function class and the
distribution class separately, and show that virtually every such combination
yields an efficient algorithm in our model. Our methods include a randomized
reduction to classification noise and an application of Le Cam's method to
obtain robust learning algorithms.
