  The problem of developing binary classifiers from positive and unlabeled data
is often encountered in machine learning. A common requirement in this setting
is to approximate posterior probabilities of positive and negative classes for
a previously unseen data point. This problem can be decomposed into two steps:
(i) the development of accurate predictors that discriminate between positive
and unlabeled data, and (ii) the accurate estimation of the prior probabilities
of positive and negative examples. In this work we primarily focus on the
latter subproblem. We study nonparametric class prior estimation and formulate
this problem as an estimation of mixing proportions in two-component mixture
models, given a sample from one of the components and another sample from the
mixture itself. We show that estimation of mixing proportions is generally
ill-defined and propose a canonical form to obtain identifiability while
maintaining the flexibility to model any distribution. We use insights from
this theory to elucidate the optimization surface of the class priors and
propose an algorithm for estimating them. To address the problems of
high-dimensional density estimation, we provide practical transformations to
low-dimensional spaces that preserve class priors. Finally, we demonstrate the
efficacy of our method on univariate and multivariate data.
