  Under label shift, the label distribution p(y) might change but the
class-conditional distributions p(x|y) do not. There are two dominant
approaches for estimating the label marginal. BBSE, a moment-matching approach
based on confusion matrices, is provably consistent and provides interpretable
error bounds. However, a maximum likelihood estimation approach, which we call
MLLS, dominates empirically. In this paper, we present a unified view of the
two methods and the first theoretical characterization of MLLS. Our
contributions include (i) consistency conditions for MLLS, which include
calibration of the classifier and a confusion matrix invertibility condition
that BBSE also requires; (ii) a unified framework, casting BBSE as roughly
equivalent to MLLS for a particular choice of calibration method; and (iii) a
decomposition of MLLS's finite-sample error into terms reflecting
miscalibration and estimation error. Our analysis attributes BBSE's statistical
inefficiency to a loss of information due to coarse calibration. Experiments on
synthetic data, MNIST, and CIFAR10 support our findings.
