  Given only positive examples and unlabeled examples (from both positive and
negative classes), we might hope nevertheless to estimate an accurate
positive-versus-negative classifier. Formally, this task is broken down into
two subtasks: (i) Mixture Proportion Estimation (MPE) -- determining the
fraction of positive examples in the unlabeled data; and (ii) PU-learning --
given such an estimate, learning the desired positive-versus-negative
classifier. Unfortunately, classical methods for both problems break down in
high-dimensional settings. Meanwhile, recently proposed heuristics lack
theoretical coherence and depend precariously on hyperparameter tuning. In this
paper, we propose two simple techniques: Best Bin Estimation (BBE) (for MPE);
and Conditional Value Ignoring Risk (CVIR), a simple objective for PU-learning.
Both methods dominate previous approaches empirically, and for BBE, we
establish formal guarantees that hold whenever we can train a model to cleanly
separate out a small subset of positive examples. Our final algorithm
(TED)$^n$, alternates between the two procedures, significantly improving both
our mixture proportion estimator and classifier
