  Weakly-supervised learning is a paradigm for alleviating the scarcity of
labeled data by leveraging lower-quality but larger-scale supervision signals.
While existing work mainly focuses on utilizing a certain type of weak
supervision, we present a probabilistic framework, learning from indirect
observations, for learning from a wide range of weak supervision in real-world
problems, e.g., noisy labels, complementary labels and coarse-grained labels.
We propose a general method based on the maximum likelihood principle, which
has desirable theoretical properties and can be straightforwardly implemented
for deep neural networks. Concretely, a discriminative model for the true
target is used for modeling the indirect observation, which is a random
variable entirely depending on the true target stochastically or
deterministically. Then, maximizing the likelihood given indirect observations
leads to an estimator of the true target implicitly. Comprehensive experiments
for two novel problem settings --- learning from multiclass label proportions
and learning from coarse-grained labels, illustrate practical usefulness of our
method and demonstrate how to integrate various sources of weak supervision.
