  We propose a method for jointly inferring labels across a collection of data
samples, where each sample consists of an observation and a prior belief about
the label. By implicitly assuming the existence of a generative model for which
a differentiable predictor is the posterior, we derive a training objective
that allows learning under weak beliefs. This formulation unifies various
machine learning settings; the weak beliefs can come in the form of noisy or
incomplete labels, likelihoods given by a different prediction mechanism on
auxiliary input, or common-sense priors reflecting knowledge about the
structure of the problem at hand. We demonstrate the proposed algorithms on
diverse problems: classification with negative training examples, learning from
rankings, weakly and self-supervised aerial imagery segmentation,
co-segmentation of video frames, and coarsely supervised text classification.
